Per Hammer Jonas Brunskog
email: jonas.brunskog@acoustics.lth.se

The Interaction Between the ISO Tapping Machine and Lightweight Floors

The ISO standard tapping machine, used as an excitation source in rating the impact-sound level of a floor structure, interacts with the floor structure during the hammer impact. Expressions for the force spectrum due to the impact are presented. The 6 dB difference at low frequencies of the force spectrum, evident in measurements, and reasons for it, are discussed. The interaction is investigated by use both of simplified lumped models and arbitrary frequency-dependent models. Local effects due to indentation near the point of impact and to global effects due to stiffeners are included in the description of the mobility involved. Numerical results are presented, where it is concluded that both the local and the global effects of the driving-point mobility are important in describing the force spectrum caused by the interaction between the tapping machine and the complex floor structure.

Introduction

The use of lightweight building techniques has increased during the last few years. It is well-known, however, that structures of this type often have poor impact-sound insulation. A prediction model is an important tool in developing structures that have acceptable insulation characteristics and in explaining how they function. A model for predicting impact noise can be said to consist of three parts: the excitation, the system and the response. The present paper focuses on the excitation part. Point impact excitation can be caused by e.g. footsteps or the impact of dropped items. Excitation can also be prodused by the ISO standard tapping machine [START_REF]Acoustics -Measurement of sound insulation in buildings and building elements -part 6: Laboratory measurments of impact sound insulation of floors[END_REF], as dealt with in the present paper. Thus, the paper aims at deriving expressions for the force spectrum produced by the impact of a hammer on a lightweight floor.

The system, or more specifically the representation of the floor system in question, is important not only as a transfer part from excitation to response, but also of deriving the impact force. It is important, therefore, that attention be directed at the interaction between the hammer and the floor. The system and the response are examined in greater detail in [START_REF] Brunskog | Prediction model for the impact sound level of lightweight floors[END_REF], in which measurements are also compared with predictions derived from a model. A comprehensive survey of the literature on prediction model approaches is presented in [START_REF] Brunskog | Prediction models of impact sound insulation on timber floor structures; a literature survay[END_REF].

The ISO standard tapping machine can be used as an excitation source for rating the impact-sound level of a floor structure. Although the machine provides no genuine simulation of real footsteps, the test results obtained yield valuable information concerning the dynamic behaviour of the floor. If the description of the tapping machine as a source is sufficiently thorough and precise, it may be possible sometime in the future to solve the problem of the correlation between sound disturbances by footsteps and the impact sound level produced by the machine.

Cremer has derived the impact noise level caused by a tapping machine for homogenous structures of high impedance, the results being summarised by Cremer and Heckl [4, pp. 269-271, 333-339]. In dealing with a bare slab Cremer makes use of momentum calculations that assume there to be a perfect elastic impact (described in section 2 of the present paper) the results are quit satisfactory. However, when taking into account the effects of a resilient floor covering, Cremer's description of the problem implies that the hammer becomes stuck to the floor after impact. This leads to resonant behaviour that is not to be found in measurements according to the ISO standard. In fact, the hammer rebounds after impact, there being only an initial positive force pulse present, a matter investigated by Lindblad [START_REF] Lindblad | Impact sound characteristics of resilient floor coverings. a study on linear and nonlinear dissipative compliance[END_REF]. Also, regarding linear excitation Lindblad considered the effect of an energy-consuming part of the deformation, conceived as a resistance in series with a spring to represent the resilient covering. The resistance can be due to local material damping or, as in the present study, to energy being transported within the plate away from the region of impact. For heavy slabs, as considered in references [START_REF] Cremer | Structure-borne sound[END_REF][START_REF] Lindblad | Impact sound characteristics of resilient floor coverings. a study on linear and nonlinear dissipative compliance[END_REF][START_REF]Vér: Impact noise isolation of composite floors[END_REF], the resilient part is due to the floor covering. Lindblad's major interest, however, was in the non-linear behaviour of coverings that interact with the hammer. Vér [START_REF]Vér: Impact noise isolation of composite floors[END_REF] derived a complete and accurate descrip-tion of the force spectrum and the impact noise level a tapping machine produces on hard surfaces, including the rebound. The improvement in insulation achieved by use of an elastic surface layer (floor covering) or of floating floors with surfaces of high impedance is also considered.

A lightweight floor structure can usually not be regarded as a homogenous structure of high impedance. It consists of thin plates of wood, chipboard, gypsum, or whatever, and is reinforced by joist stiffeners. Thus, it cannot be assumed that the force spectrum in [START_REF] Cremer | Structure-borne sound[END_REF][START_REF] Lindblad | Impact sound characteristics of resilient floor coverings. a study on linear and nonlinear dissipative compliance[END_REF][START_REF]Vér: Impact noise isolation of composite floors[END_REF] is applicable generally. Nevertheless, in more recent papers on impactsound insulation, such as Gerretsen [START_REF] Gerretsen | Calculation of airborne and impact sound insulation between dwellings[END_REF], the momentum model of Cremer and Heckl [START_REF] Cremer | Structure-borne sound[END_REF] has continued to be used, also for non-homogenous or lightweight floors. Thus, it is common to assume the force spectrum to be invariant with respect to the excitation system (i.e. of its being a linear source with infinite source mobility). In the present paper no such assumption is made. In another recent paper, by Scholl et al. [START_REF] Scholl | Impact sound insulation of timber floors: Interaction between source, floor coverings and load bearing floor[END_REF], the interaction between the source, the floor covering and the floor structure is considered. However, the floor structure is represented by the mass of the structure, not taking into account that the driving-point impedance of a plate on average has the characteristics of a resistance (i.e. 8 p m 00 B), and in general is complex. Thus, no energy (or momentum) consuming part is included in the model used by Scholl. Measurements, such as those of Hall's [START_REF] Hall | The standard impact test and thin floating floor constructions[END_REF], indicates there to be a 6 dB gap at low frequencies between the force (or acceleration) spectrum of a hammer impacting on a high impedance surface such as concrete and its impacting on a low impedance surface such as an mdf-board. This can easily be explained in terms of simple momentum consideration in a manner comparable with the case to which Cremer's [START_REF] Cremer | Structure-borne sound[END_REF] calculations but letting the hammer be stuck to the plate, as will be taken up in section 2.

The organisation of the paper is as follows: In order to obtain a force spectrum of a form suitable for the solution technique applied to the system described in [START_REF] Brunskog | Prediction model for the impact sound level of lightweight floors[END_REF], the findings reported in [START_REF] Cremer | Structure-borne sound[END_REF][START_REF] Lindblad | Impact sound characteristics of resilient floor coverings. a study on linear and nonlinear dissipative compliance[END_REF][START_REF]Vér: Impact noise isolation of composite floors[END_REF] are integrated and reanalysed in sections 2 and 3, several numerical examples being provided there. A system of arbitrary frequencydependent driving-point mobilities is then added in section 4, in which different causes for the frequency-dependent driving-point mobilities are discussed. The procedure for evaluating these mobilities numerically is described in section 5. The numerical results are presented in section 6, the conclusions being summarized, finally, in section 7.

The present analysis is based on linear theory implying that the contact area during impact is constant, as discussed in section 3.2. The displacement of the receiver structure is in addition small, and therefore assumed to be within the linear range. It should be noted, however, that a direct linear-system analysis is not applicable since the system is changed when the object producing the impact leaves the structure with which it has had contact. Thus, the boundary condition between the object and the structure is rather being an inequality than an equation, and the situation is not time-invariant.

The hammer impact force in the time and the frequency domain

The ISO standard tapping machine [START_REF]Acoustics -Measurement of sound insulation in buildings and building elements -part 6: Laboratory measurments of impact sound insulation of floors[END_REF] consists of five hammers spaced equally along a line 40 cm in length.

As an approximation, it is assumed here, however, that all hammer impacts act on the same position. Each hammer has a mass of M = 0:5 kg and it is dropped from a height h of 4 cm. The hammers strike the floor with a rate of f r = 10 times per second, giving a repetition time of T r = 1=f r = 0:1 s. Consider initially, however, a single hammer impact with the force time history f 1 (t). The Fourier spectrum of this force pulse is F 1 (f) = F t ff 1 (t)g, F t f g being the Fourier transform operator (time to frequency). In the present paper, the word spectrum is used to denote a function in the (Fourier) frequency domain. The excitation caused by the tapping machine can be regarded as an array of separate force pulses f 1 (t). The time history of the repeated force f R (t

) is thus f R (t) = 1 X n=;1 f 1 (t ; nT r ): (1)
This time history is a periodic signal. Accordingly, it can be represented by a Fourier series, e.g. by the two-sided complex Fourier series

f R (t) = 1 X n=;1
F n e i2 nt=Tr :

(

The signal is represented by a Fourier series consisting of an infinite number of discrete frequency components of amplitude F n . The Fourier spectrum for the signal is the tonal spectrum

F R (f) = F t ff R (t)g = 1 X n=;1 F n (f ; nf r ) (3)
where F R (f) denotes the spectrum of the repeated signal.

It was used in [START_REF] Brunskog | Prediction model for the impact sound level of lightweight floors[END_REF] as the excitation force of the system, whereas denotes the Dirac delta function. Each amplitude is given by

F n = 1 T r Z Tr 0 f 1 (t)e ;i2 nt=Tr dt (4)
where, as indicated above, f 1 (t) is the force time history of a single hammer impact. This integral is identical with the Fourier transform of the individual force pulse except for the factor 1=T r . Thus, for the tapping machine the repeated force components F n is 10 s ;1 times the force spectrum for a single impact. If a trigonometric Fourier series is used instead, each sine or cosine component is twice that in (4). This is due to the two-sided representation described in [START_REF] Brunskog | Prediction model for the impact sound level of lightweight floors[END_REF], which is the most suitable representation in the present situation, since the force spectrum is used in a two-sided model, i.e. one in which there is assumed to be an e iwt dependence. Cremer [START_REF] Cremer | Structure-borne sound[END_REF] and Vér [START_REF]Vér: Impact noise isolation of composite floors[END_REF] used a one-sided rep-resentation, together with a RMS and a (third) octave band procedure.

For low-frequency components the force pulse is usually short compared with the period of interest. Accordingly, the Fourier amplitude of the force pulse train during the effective interval of the force pulse, exp(;i2 nt=T r ) 1, can be approximated by

F n j f!0 = 1 T r Z Tr 0 f 1 (t)dt (5)
which is the mechanical impulse divided by T r , the mechanical impulse being equal to the change in momentum. The hammer hits the slab with a velocity 0 = (2gh) 1=2 = 0 :886 m/s. If the impact is purely elastic, the momentum of the hammer after impact is equal in magnitude to that prior to impact but is of the opposite sign, the hammer lifting from the slab with the velocity 0 .

Thus, F n j f!0 = 2 M 0 =T r = 8 :859 N (where for a single force pulse the corresponding low-frequency asymptote is F 1 j f!0 = 2M 0 ). Since this is the highest possible low-frequency asymptote of the impact spectrum, it is the maximum value for the magnitude of the spectrum. At the other extreme, if the impact is so damped that the entire momentum is dissipated during impact, the hammer does not rebound. The mechanical impulse is then

F n j f!0 = M 0 =T r = 4 :430 N (F 1 j f!0 = M 0 ). This
is the lowest possible low-frequency asymptote of the impact spectrum. Thus, these two cases constitute the upper and lower bounds of the low-frequency asymptote of the force spectrum, which represents a span of 4.429 N, or 6 dB. As indicated above, this span can be clearly seen in measurements that have been made [START_REF] Hall | The standard impact test and thin floating floor constructions[END_REF].

Impact force and lumped system

In order to include cases between the two extremes just referred to, a lumped model can be employed, one that provides a somewhat more realistic description than that of momentum consideration does. The model and the solution arrived at are taken from Lindblad [START_REF] Lindblad | Impact sound characteristics of resilient floor coverings. a study on linear and nonlinear dissipative compliance[END_REF], whereas the physical situation and the analysis are new. A treatment of general frequency-dependent impedances is found in section 4.

A single slab, interaction between hammer and floor

A lumped model of the impact of the hammer on a single slab floor is shown in Figure 1. The floor consists of a resilient part and an energy consuming part, represented by a spring with stiffness K and by a dashpot with resistance R, respectively. The physical meaning of the two components is discussed in section 3.2.

When the hammer has reached the slab, the differential equation for the system assumed is

M @ 2 K =@t 2 = K( K ; R ) K( R ; K ) = R@ R =@t (6) M K R υ 0 ξ K ξ R Figure 1. Model of hammer impact.
where K and R are displacements. Under the initial condition 0 , and assuming frequency independent K and R, the solution is found to be [START_REF] Lindblad | Impact sound characteristics of resilient floor coverings. a study on linear and nonlinear dissipative compliance[END_REF]

f 1 (t) = ( 0 K sinh( oct) oc e ;K t = 2R K M 4R 2 0 K sin( uct) uc e ;K t = 2R K M<4R 2 (7)
which of these applies depending on whether the oscillation is overcritical or not, where

oc = p (K=2R) 2 ; K=M uc = p K=M ; (K=2R) 2
are the overcritical and the undercritical angular frequency, respectively. In Lindblad [START_REF] Lindblad | Impact sound characteristics of resilient floor coverings. a study on linear and nonlinear dissipative compliance[END_REF] the first of these is denoted as the 'stuck to the floor' case and the latter as the 'rebound' case. If an overcritical oscillation is present, a numerically more appropriate form would be to write the expression in terms of exponential functions instead of hyperbolics combined with exponentials. For R ! 1 , or if R = 0 , equation ( 7) is replaced by

f 1 (t) = 0 K sin( p K=M t )= p K=M (8)
which is an undamped oscillation in which f ud cut = 1=(2 )(K=M) 1=2 is the undamped natural frequency (and cutoff frequency). The force starts at zero at the moment the hammer hits the slab, and increases to a maximum, at which point the hammer reaches its maximum depth, the hammer then starting to return and the force decreasing.

As the force reaches the zero crossing at t cut = 1 =(2f cut ), the hammer rebounds, takes off from the slab and is picked up by a catching mechanism (an eccentric cam). Thus, the force is zero after this moment,

f 1 (t) = ( 0K p K=M sin( p K=M t ) 0 < t < 1 2fcut 0 else:
The same conditions hold for the damped oscillations described in [START_REF] Gerretsen | Calculation of airborne and impact sound insulation between dwellings[END_REF]. In the overcritical case the force will never completely be zero. However, the force still decreases rapidly after reaching the maximum and is approximately zero at T r . Each amplitude in the tonal spectrum of the ISO tapping machine is given then by

F n = 1 T r Z Tr 1 0 f 1 (t)e ;i2 nt=Tr dt (9)
in the case of overcritical damping, and by

F n = 1 T r Z tcut 0 f 1 (t)e ;i2
nt=Tr dt [START_REF] Timoshenko | Theory of elasticity[END_REF] in the case of undercritical damping. These integrals can be expressed in closed form, but are likewise well suited for numerical integration. The Fourier transforms of a single impact are given below, where the cutoff frequencies are also determined. For the over-critical case, the inequality K=2R > oc holds, giving a Fourier transform over time to angular frequency ! = 2 fof equation ( 7),

F 1 oc = F t ff 1 (t)j oc (t)g = 0 K M K ; ! 2 M + i !KM=R (11)
where (t) is the unit step function. The low-frequency asymptote of ( 11) is F 1 oc j f!0 = 0 M as expected. In the undercritical case, a Fourier transform over time of equation ( 7) yields, taking into account the time interval of interest

F 1 uc = F t ff 1 (t)j uc ; (t) ; (t ; t cut ) g = 0 K M (1 + e ;tcut (i!+K=2R) ) K ; ! 2 M + i !KM=R (12)
where

t cut = = uc = 1 =2f cut
is the time of zero-crossing. The low-frequency asymptote is

F 1 uc j f!0 = 0 M ; 1 + e ;K t cut=2R
which has two extremes depending on the resistance R,

F 1 uc = 2 0 M f ! 0 a n d R ! 1 0 M f ! 0 a n d R ! 1 2 p K M
which agrees with the asymptotes schematically derived in section 2 on the basis of the mechanical impulse and the change in momentum.

In both the overcritical and the undercritical case, the Fourier series components are then found to be F n = F 1 (nf r )f r , and the complex cutoff angular frequency (i.e.

the poles) is

! cut = i K=2R p K=M ; K 2 =(2R) 2 (13)
If the negative sign is chosen, the absolute value of the complex cutoff angular frequency yields the point of interest on the real axis, (14) The cutoff frequency then is

j! cut j = ( K 2R ; q K 2 (2R) 2 ; K M K M 4R 2 q K M K M<4R 2
f cut = j! cut j=2 (15a)
whereas in the undercritical case the undamped cutoff frequency,

f ud cut = 1 =(2 ) p K=M (15b)
is employed, this being the frequency at which the phase equals ; =2.

Choice of the frequency independent stiffness and resistance

In Figure 1 the impedance at the position where the hammer hits the floor is represented by a spring and a dashpot in series. Thus, a suitable stiffness K and a resistance R need to be found in order to achieve an adequate approximation of what occurs at impact. The resilient part is often the result of there being an elastic surface layer on an otherwise bending stiff slab. The stiffness of the elastic layer is then K = E A h =d, c.f. Vér [START_REF]Vér: Impact noise isolation of composite floors[END_REF], where E is Young's modulus, d is the thickness of the elastic layer, and A h is the area of the hammer. The resistance is then related to the local dissipation, R = (K M ) 1=2 , being the loss factor for the material. In the lightweight floor structures considered in this paper, however, the hammer hits a rather thin plate made of gypsum or of wooden material. It can thus be assumed that the resilient part is due to local deformation of the plate, and the resistive part to energy transportation within the plate. As a first approximation, the stiffness of the local deformation can be found, as in [START_REF] Timoshenko | Theory of elasticity[END_REF][START_REF] Petersson | Concentrated excitaion of structures[END_REF], by

K = 2 GD h =(1 ; ) = E D h =(1 ; 2 ) ( 16
)
where G is the shear modulus, is Poisson's ratio and D h = 2 ( A h = ) 1=2 is the diameter of the hammer. The local stiffness here is found for a static deformation caused by a rigid stamp on a semi-infinite elastic solid, the socalled Bossinesq deformation. If the area of contact between the hammer and the floor can be regarded as involving contact between two elastic bodies of different radii, a geometric non-linearity will occur, a so-called Hertz deformation [START_REF] Timoshenko | Theory of elasticity[END_REF]. The hammers of the tapping machine are actually not entirely flat, but since after a few impacts the material in the impact zone becomes somewhat plastically deformed by the hammer it is assumed that the Hertz deformation effect will not be dominant in the steady-state vibrational phase. The resistance is taken to be the real driving-point impedance of a thin plate, R = 8 p m 00 B [START_REF]MATLAB Reference guide[END_REF] its being taken into account that energy is transported away from the excitation point by bending wave motion. Here m 00 is the mass per unit area and B = E I 0 is the bending stiffness of the plate.

Both of these lumped parameters are frequency independent, a necessary condition for the solution technique described in section 3.1.

Numerical examples

Various numerical examples will be presented now to illustrate certain features of the lumped two-parameter description of the impact force of the tapping machine. The examples will come from a specific combination of R and K, where R = 2 10 3 Ns/m and K = 4 10 6 N/m. These values correspond roughly to an infinite 22 mm thick wooden plate, use being made of equations ( 16) and [START_REF]MATLAB Reference guide[END_REF], respectively. The parameters are varied one at the time.

In Figures 2 to 4 the stiffness is varied as K = f5 1 0 2 0 4 0 8 0 g 10 6 N/m. Figure 2 shows the time history of the impact force. The first three impact forces are undercritical, as indicated by the dashed line (---), since they should oscillate if uninterrupted. The increase in stiffness results in a narrower pulse, and thus a somewhat broader frequency range, as can be seen in Figure 3. Figure 3 shows the magnitude of the force spectrum, as derived from equation [START_REF] Petersson | Concentrated excitaion of structures[END_REF] or [START_REF] Gudmundsson | Sound insulation improvment of floating floors. A study of parameters[END_REF]. The undamped cutoff frequency (15b) is indicated by circles (o) and the actual cutoff frequency (15a) by pluses (+). The extremes of the low-frequency asymptotes are indicated by two stars on the ordinata ( ). The undercritical behaviour of the first three pulses manifests itself as oscillation in the highfrequency range. The overcritical force pulses are close to the lower extreme of the low-frequency asymptotes.

The behaviour of the cutoff frequency when the stiffness increases can be described as follows: The cutoff Thereafter the cutoff frequency decreases as the stiffness increases further. The undamped cutoff frequency always increases with increasing stiffness.

Figure 4 shows the phase of the force spectrum. The undamped cutoff frequency (15b) is marked by circles (o). As a reference to the phases, the time of impact is taken, giving a low-frequency asymptote of zero.

In Figure 5 to 7 the resistance is varied as R = f0:5 1 2 4 8 g 10 3 Ns/m. Figure 5 shows the time history

10 100 1k 5k -3/2 -1 -1/2 0 ∠ F n , ∠ F 1 /T R [rad] f [Hz] × π
Increasing resistance of the impact force. The two final impact forces are undercritical. The dashed line (---) indicates how they would have continued if uninterrupted. An increase in resistance results in the pulse becoming narrower, and thus in the frequency range becoming somewhat broader, as can be seen in Figure 6. The undamped cutoff frequency remains unchanged, however.

Figure 6 displays the magnitude of the force spectrum, as derived from equation [START_REF] Petersson | Concentrated excitaion of structures[END_REF] or [START_REF] Gudmundsson | Sound insulation improvment of floating floors. A study of parameters[END_REF]. The undamped cutoff frequency (15b) is indicated by circles (o) and the actual cutoff frequency (15a) by pluses (+). The extremes of the low-frequency asymptote are indicated by two stars on the ordinata ( ). The last spectrum (R = 8 10 3 Ns/m) is not much affected by dissipation, the low-frequency asymptote consequently having increased. For the overcritical pulses, the spectrum starts to decrease at the actual (lower) cutoff frequency, not at the undamped one.

Figure 7 shows the phase of the force spectrum involved. The reference to the phases is taken at the time of impact, giving a low-frequency asymptote of zero. The undamped cutoff frequency (15b) is indicated by circles (o).

To conclude: increasing the stiffness gives a lower low-frequency asymptote and a higher cutoff frequency, whereas increasing the resistance gives a higher lowfrequency asymptote and an unchanged undamped cutoff frequency (15b) and a lower actual cutoff frequency (15a).

Floating floors, a discussion

A floating floor can be described with two infinite plates connected by a resilient layer [START_REF] Cremer | Structure-borne sound[END_REF][START_REF] Gudmundsson | Sound insulation improvment of floating floors. A study of parameters[END_REF]. To simplify the situation, it can be assumed that the plates are thin and that the resilient layer is point-reacting and massless. To simplify the situation still further, two different cases can be distinguished such that in the first case the impedance of the excited plate is so high that the impact situation is the same as described in section 3.1-3.2, whereas in the second case the impedance of the excited plate is so low that the secondary (non-excited) plate can be regarded as rigid. In this second case the plate system can be simplified to a Winkler foundation, involving a plate resting on a locally reacting resilient layer. The bending wave-number is k 4 B = (m 00 ! 2 ; K)=B, where m 00 is the mass per unit area of the plate, K is the stiffness of the foundation and B is the bending stiffness of the plate. The driving-point mobility then is

Y = ! ;8B 1=2 i p K ; m 00 ! 2
where the square-root in the denominator yield as a consequence that no lumped model can be used, even for this simple case. Thus, a more general description is needed.

Impact force and the general system

In section 3 the impact force spectrum was derived for frequency-independent parameters K and R in a mechanical series. The spectrum was given explicit expressions [START_REF] Petersson | Concentrated excitaion of structures[END_REF][START_REF] Gudmundsson | Sound insulation improvment of floating floors. A study of parameters[END_REF]. However, the floor system cannot generally be described in terms of frequency-independent parameters. The lumped description of the driving-point impedance/mobility is basically an ad hoc approximation of the actual situation. A more accurate description would be to calculate the driving-point mobility from the system description. The driving-point mobility is defined as the complex ratio of velocity to applied force, where the velocity is measured at the point of application of the force, c.f. [START_REF] Morfey | Dictionary of acoustics[END_REF].

Since the methods employed in section 3 cannot be used for frequency-dependent mobility, however, an approach to finding the force spectrum for an arbitrary driving-point mobility needs to be found. A suitable approach is to solve the differential equations in the frequency domain, inverse transform the result obtained in order to find the time of rebound, and to then transform the remaining force into the frequency range.

The interaction between the hammer and the floor

Figure 8 shows the generalised impact situation and the procedure employed. The mobility is frequency-dependent, implying that the equations of motion can easily be solved in the frequency domain. On the other hand, the impact history is interrupted after the first zero crossing of the force. Thus the system is not time-invariant and therefore is best treated in the time domain. To deal with this problem, consider

υ 0 M Z b) f 1 (t) M Z c) F 0 δ(t) M d) F 0 δ(t) f 1 ´(t) Figure 8. Modified impact description, impedance Z = 1 = Y .
the impedance Z. The entire system is driven by a force F 0 (t). This modified form of the problem is both linear and time-invariant. The equation of motion and the floor reaction force, if taken in the frequency domain, are

F 0 ; F 0 1 = i !Mv F 0 1 = v=Y (18)
where v = F t f (t)g is the velocity spectrum of the floor and F 0 1 is the spectrum of the continuing impact force, i.e. the floor reaction force between the mass and the impedance, as shown in Figure 8 c-d), F 0 1 = F t ff 0 1 (t)g.

One solves then for F 0 1 and v, F 0 1 = F 0 =(1 + i!MY)

(19a) v = Y F 0 1 : (19b)
The magnitude of the force F 0 needs to be selected so that the velocity at t = 0 + equals the velocity of the falling hammer. At t = 0 ; the velocity should be zero, (0 + ) = 0 (0 ; ) = 0 :

In evaluating (0), however, account needs to be taken of the fact that the Dirac (t) that excites the floor at time t = 0 is only at half inside the infinitesimal region from t = 0 to t = 0 + . This is compensated for if a value halfway between (0 + ) and (0 ; ) is employed, (0)j Dirac = 0 =2:

(21)
The velocity of the floor at t = 0 is evaluated as the integral over all the frequencies,

(0) = 1 2 Z 1 ;1 v(!)d! = F 0 2 Z 1 ;1 d! i!M+ 1 = Y(!) F 0 I 0 (22)
where equation (19b) is used in next-to-the-last equality, and the last equality is the definition of the integral I 0 .

One then inserts ( 21) in ( 22) to obtain the magnitude of the driving force in the modified system,

F 0 = 0 =2I 0 (23)
where the integral I 0 (mostly) needs to be calculated numerically. Equation (19a) then becomes

F 0 1 = 0 =2I 0 1 + i !MY(!) : (24)
The time history of this force can be found by means of the inverse Fourier transform, f 0 1 (t) = F ;1 t fF 0 1 g, implemented numerically as a fast digital inverse transform. The moment in time of the first zero crossing of the force is obtained then as

t cut = m i n ftjt > 0 f 0 1 (t) = 0 g: (25)
The actual, interrupted, excitation impact force then is

f 1 (t) = f 0 1 (t) (t cut ; t) (26)
the corresponding force spectrum being found by means of a Fourier transform F 1 = F t ff 1 (t)g, implemented numerically as a fast digital transform. The Fourier series components of ( 4) are F n = F 1 (nf r )f r , where f r = 1 =T r is the repetition frequency of the tapping machine.

The driving-point mobility

In the lumped system description presented in section 3, which made use of frequency-independent components, both the stiffness and the resistance were important for the solution. For a general mobility, this implies that both the real and the imaginary part of the mobility are important.

For an infinite homogenous plate, the imaginary part of the mobility, or the finite stiffness, can only be due to local effects. In section 3, the stiffness was chosen to be the stiffness due to deformation near the impact zone on a semi-infinite elastic solid, the Bossinesq expression [START_REF] Brunskog | Rigid indenter excitation of plates[END_REF]. It would be more realistic to determine the local stiffness of a plate of finite thickness, i.e. to treat the plate as an elastic continuum. The force excites the plate asymmetrically on its upper surface. Thus, the excitation force produces a complicated displacement field under and near the point of excitation.

However, the description of the elastic continuum has too high a level of complexity to be appropriate for the entire system, including the reinforcing beams, for example. To this end, it is better to use thin plate theory, assuming plane sections to remain plane in the plate, which implies the excitation force to be constrained to produce a uniform displacement field in the thickness direction. Thus, no local deformation can be attained in terms of thin plate theory.

A heuristic description of the driving-point mobility could be to combine the mobility as determined for the global system, Y G , with the mobility as determined for the detailed description near the excitation point, Y L . Note that both Y G and Y L are in general complex. The parts of the mobilities that overlap need to be subtracted, i.e. the mobility of an infinitely thin plate. The total mobility is expressed as a correction of the global mobility,

Y = Y G + 4Y 4Y = Y L ; 1=8 p m 00 B (27)
where m 00 is the mass per unit area and B is the bending stiffness of the excited plate. This is quite an elaborate description of the driving-point mobility. The mobilities Y L and Y G need to be determined then.

One can also use measured mobilities as an alternative to the theoretically derived mobilities, provided the frequency resolution is sufficiently high.

Local effects on the mobility

The mobility Y L due to local effects needs to be determined, the global parts of the system being excluded. The mobility of a plate of finite thickness and infinite extent excited on its upper surface by an indenter, is described in the literature. The indenter is assumed to be circular, weightless and stiff as compared with the plate, and to be small as compared with the wavelengths of the bending and quasi-longitudinal waves of the plate. A rigid indenter, such as the case of a metal hammer acting on a wooden or a gypsum plate, can be assumed to provide a better approximation of the actual situation than a soft indenter would. Use of a rigid indenter is also more reasonable than use of a soft one (assuming the pressure distribution to be uniform), since it allows the pressure distribution under the indenter to change as the frequency increases.

In all analyses describing the motion of 'thin' structures, such as in the Kirchoff and the Mindlin theories, it is assumed that the two sides of the structure have exactly the same displacement at each point. This is an approximation, and both additional weakness and inertia effects can occur. Thus, simplified 'thin' theories are not sufficient for the case at hand.

More detailed three-dimensional analyses have been carried out, such as by Ljunggren [START_REF] Ljunggren | Generation of waves in an elastic plate by a vertical force and by a moment in the vertical plane[END_REF] for a rigid indenter and by Heckl [START_REF] Heckl | K orperschall ubertragung bei homogenen platten beliebiger dicke[END_REF] for a soft indenter. Petersson and Heckl [START_REF] Petersson | Concentrated excitaion of structures[END_REF] have investigated the influence of different choices of pressure distribution. The boundary value problem is simpler mathematically in the case of a soft indenter than of a rigid one. Therefore, by assuming a pressure distribution under the indenter, Ljunggren [START_REF] Ljunggren | Generation of waves in an elastic plate by a vertical force and by a moment in the vertical plane[END_REF], and Petersson and Heckl [START_REF] Petersson | Concentrated excitaion of structures[END_REF] (when dealing with a rigid indenter) avoided the problems a rigid indenter involves. The pressure distribution taken was that of a rigid indenter statically loading an elastic semi-infinity. Since there is no guarantee that this assumption actually results in a uniform displacement under the indenter, such a case can be designated as quasirigid.

In a paper by the authors [START_REF] Brunskog | Rigid indenter excitation of plates[END_REF], the pressure distribution at the interface between the indenter and the plate was determined by use of a variational formulation. The expressions obtained are approximations, although the choice of these is an optimal one. The mobility described in [START_REF] Brunskog | Rigid indenter excitation of plates[END_REF] is written as

Y L = 1 2 R 2 I a I c ; I 2 b I c ; I b + I a =4 (28)
where

I a = Z 1 0 sin 2 (Rk r ) k r A(k r )dk r (29)
I b = Z 1 0 J 1 (k r R) s i n ( Rk r ) k r A(k r )dk r (30)
I c = Z 1 0 J 2 1 (k r R) k r A(k r )dk r (31)
and where A(k r ) is the admittance, used in all the references [START_REF] Petersson | Concentrated excitaion of structures[END_REF][START_REF] Ljunggren | Generation of waves in an elastic plate by a vertical force and by a moment in the vertical plane[END_REF][START_REF] Heckl | K orperschall ubertragung bei homogenen platten beliebiger dicke[END_REF][START_REF] Brunskog | Rigid indenter excitation of plates[END_REF].

It should be pointed out that only the imaginary part of Y L can be said to be due to local deformation, the real part being due to the rest of the system as well, and it is in a broad frequency range close to 1=8 p m 00 B.

Global effects on the mobility

The mobility Y G due to global considerations needs to be determined, local effects near the excitation zone being excluded. Both the real and the imaginary parts of the driving-point mobility of the floor structure may be due to global effects. An infinite thin plate has a real mobility, whereas in a finite plate, if no damping is included, the mobility is entirely imaginary. Thus, an infinite plate has only a resistance part in the mobility, whereas a finite plate has only stiffness and mass parts in the mobility. In lightweight floors consisting of plates reinforced by beams, the mobility has an imaginary part, also in the case of infinite systems. In [START_REF] Brunskog | Prediction model for the impact sound level of lightweight floors[END_REF], a typical lightweight floor system is described by use of a spatial Fourier transform method. The drivingpoint mobility due to global effects can be derived using the same strategy. Assume that the transformed displacement field is found in a way similar to that presented in [START_REF] Brunskog | Prediction model for the impact sound level of lightweight floors[END_REF]. The driving-point displacement can be found then by means of a double inverse Fourier transform in spatial coordinates,

w(x 0 y 0 ) = 1 4 Z Z 1 ;1 w( )e ;i(x0 +y0 ) d d (32)
where w is the displacement field of the excited plate, w is the spatially transformed displacement field, x 0 and y 0 are the coordinates of excitation, and and are the transform wavenumbers. An account of how to simplify and reduce the integrals in (32) to be suited for numerical integration is presented in the Appendix. A simplified floor system is also described there.

The driving-point mobility due to global considerations is defined as

Y G = v(x 0 y 0 )=F 1 = i !w(x 0 y 0 )=F 1 (33)
How the driving-point mobility is determined is not crucial, however, for the methods in section 4.1. One can use modal sums, FEM or any other deterministic method, as long as the frequency resolution is sufficiently high.

Numerical evaluation and programming

Since the impact force description of the general drivingpoint mobility is not given in the form of a closed expression, program coding is an important part of describing the excitation. Certain important aspects of the program developed will be taken up and be shown in a flowchart (Figure 9).

The program starts by defining the input parameters, including that the driving-point mobility Y is to be determined. Equations (28-31) are used for the local part of the mobility. The global part of the mobility can be integrated from equations (32-33), see also the Appendix. In the numerical example to be taken up, an adaptive and recursive Newton-Cotes eight-panel integration scheme, the Matlab function quad8 [START_REF]MATLAB Reference guide[END_REF], was used in the numerical evaluation. In order to speed up the calculations, a frequency resolution of 10 Hz was employed, and the frequency range being 10 Hz to 6000 Hz. The numerical 'signal theoretic' methods described in section 4.1 are in need of a better frequency resolution and of a broader frequency range, as will soon be evident. The narrower frequency range is achieved using a spline interpolation, and the extended frequency range is achieved by using the asymptotic mobilities of an infinite plate for excitation positions between beams, and of an infinite beam for excitation positions at the beams.

The magnitude of the Dirac shown in Figure 8 c needs to be calculated then, by use of ( 22) and ( 23). The spectrum of the continuing force is calculated then from (24) and numerically inverse transformed by use of the IFFT algorithm. It is important that the frequency resolution is sufficient to describe the pulse peak and detect the first zero-crossing correctly.

As an example, consider the lumped model with R = 2 10 3 ; and K = 2 10 7 , use of [START_REF] Ljunggren | Generation of waves in an elastic plate by a vertical force and by a moment in the vertical plane[END_REF] showing that t cut = 8:112 10 ;4 s. An appropriate time resolution might then be > 20 points for describing the force pulse. Thus, the time resolution should be 4t < 4:06 10 ;5 s, giving an upper frequency limit of f nyq > 12:3 10 3 Hz (the Nyquist frequency). If a frequency resolution of 4f = 2 Hz is employed, an FFT/IFFT of 16384 points is needed.

In the numerical example to be presented, the resolution data chosen were; 4t = 8:33 10 ;6 s, 4f = 3:66 Hz and f nyq = 6 0 10 3 Hz, a 16384-points FFT/IFFT being employed. Since the impact noise is often only of interest up to 5000 Hz a low-pass filter of 10 10 3 Hz was used, to prevent high-frequency terms from influencing the force spectrum.

The first zero-crossing needs to be found then, so that the actual, interrupted impact force was determined from (26). The zero-crossing was found by examining the change of sign. The exact position could not be determined since the time resolution is fixed. The force spectrum was calculated then by use of the fast digital Fourier transform FFT, the same resolution data as before being employed.

Due to the material models and numerical procedures there will be some causality problems in the procedure. If the force is non-zero before and at the time of impact t = 0 it clearly violating the causality rules. This not only produces a wrong result, but also complicates the numerical treatment since sometimes more than one zero crossing may need to be found. However, when the time and frequency resolution are good and the damping is slight, problems of this sort are held to a minimum, its being assumed in such cases that the errors involved can be disregarded.

Numerical results and discussion

A numerical example will be given to illustrate the excitation force description when a frequency dependent driving-point mobility is employed. One plate reinforced by one set of periodically spaced beams, as described in the Appendix, is used instead of the complete floor structure, so as to simplify the calculations.

The following data were used in the numerical calculations: distance between beams l = 0 :6 m, modulus of the plates E p =(1 ; 2 ) = 7 :2 10 9 Pa, Young's modulus for the framing beams E f = 9 :8 10 9 Pa, density of the beams and plates p = f = 5 0 0 k g =m 3 , thickness of the plates h = 22 10 ;3 m, and material damping = 0:03. The beams are 0.220 m in height and 0.067 m in width.

The magnitude of the driving-point mobility is shown in Figure 10, calculated there for 15 positions. These important, since these differ both in the low-frequency asymptote and at the cutoff frequency. The extremes of the low-frequency asymptotes are indicated by one star ( ) and one circle ( ) on the ordinata. One of the curves fall outside the extremes, implying the numerical procedures to not be perfect. The errors, however, are minor, the important features being clearly illustrated. It should be pointed out that for this case there are no problems concerning causality.

In Figure 12

Summary and conclusions

Findings concerning the excitation force achieved by the ISO tapping machine, as reported in the literature, have been summarised and reconsidered. A two-parameter lumped model of impact developed by Lindblad [START_REF] Lindblad | Impact sound characteristics of resilient floor coverings. a study on linear and nonlinear dissipative compliance[END_REF], was analysed. The model can only deal with frequency-independent parameters. Low-frequency asymptotes and cutoff frequencies were derived. The low-frequency asymptotes were found to have a span of 6 dB. The lumped parameters were taken as the local stiffness and the driving-point mobility of an infinite plate. On the basis of a numerical parametric study, it was concluded that increasing the stiffness gives a lower low-frequency asymptote and a higher cutoff frequency, and that increasing the resistance gives a higher low-frequency asymptote, as well as an unchanged undamped cutoff frequency and a lower actual cutoff frequency.

A description of impact force applicable to general, frequency-dependent impedances and mobilities was derived. The general force description was implemented by means of numerical integration and FFT. The mobilities may be due to local effects, to the use of thick plate theory, and/or global effects, obtained using spatial Fourier transform methods and numerical integration. From a numerical example it was concluded that both the local and the global effects are important in determining the excitation force of the ISO tapping machine on a non-homogenous lightweight floor. In order to adequately describe what occurs, the global and the local driving-point mobility has to be used and combined. The results also indicate how important it is to use an accurate and detailed system description in order to predict the impact force spectrum appropriate. The force spectrum needs to be determined on the basis of the entire driving-point mobility, that is both the real and the imaginary parts.

Figure 2 .

 2 Figure 2. Time history of hammer impact. Solid line (-) denotes the interrupted force, dashed line (---) denotes the oscillating force. The parameters are K = f5 10 20 40 80g 10 6 N/m, R = 2 10 3 Ns/m.

Figure 3 .

 3 Figure 3. Hammer impact spectrum, showing magnitudes. The parameters are K = f5 10 20 40 80g 10 6 N/m and R = 2 10 3 Ns/m.

Figure 4 .

 4 Figure 4. Hammer impact spectrum, showing the phases. The parameters are K = f5 10 20 40 80g 10 6 N/m and R = 2 10 3 Ns/m.

Figure 5 .Figure 6 .

 56 Figure 5. Hammer impact time history. Solid line (-) denotes the interrupted force, dashed line (---) denotes the oscillating force. The parameters are R = f0:5 1 2 4 8g 10 3 Ns/m and K = 4 0 10 6 N/m.

Figure 7 .

 7 Figure 7. Hammer impact spectrum: phases. The parameters are R = f0:5 1 2 4 8 g 10 3 Ns/m and K = 4 0 10 6 N/m.

Figure 8 a

 8 Figure8shows the generalised impact situation and the procedure employed. Figure8 a) presents a general model of the impact. The hammer, of mass M, strikes the floor with the velocity 0 . The floor can be described in terms of the general driving-point mobility Y (or impedance Z = 1= Y). The desired force, f 1 (t), is shown in Figure8 b).

Figure 8 c

 8 Figure 8 c), in which the mass M is now fixed on top of

Figure 9 .

 9 Figure 9. Flow chart for calculation of the force spectrum.

Figure 10 .

 10 Figure 10. Global driving-point mobility at 15 positions, solid line (-). Driving-point mobility for an infinite plane, dashed line (---). Driving-point mobility for an infinite beam, dashed-dotted line (--).

Figure 11 .

 11 Figure 11. Force spectrum, calculated according to sections 4 and 5. Mobility is taken as the global one, see section 4.4, no local weakness are present.

Figure 12 .

 12 Figure 12. Force spectrum, calculated according to 4 and 5. Mobility taken as the global one in section 4.4, but with local weakness added in accordance with section 4.3.

  the local mobility has been added to the global mobility, in accordance with equation (27), the force spectrum being calculated then. Local effects are also important; the corresponding excitation points in Figures 11 and 12 differ both in the asymptote and in the cutoff frequency. The cutoff frequency and the low-frequency asymptote are consisting less in Figure 12.

Acknowledgements

The authors thank their colleagues at the Division of Engineering Acoustics at LTH, Lund University, for the helpful suggestions and criticism they have provided, in particular Dag Holmberg. The financial support provided by Skanska Teknik AB, 'The Building and its Indoor Environment' research school at Lund university (the KK-foundation) and SBUF is also thankfully acknowledged.

Appendix

The driving-point mobility is needed for determining the force spectrum of the tapping machine. A simplified system for performing the calculations is considered, one taking account of the upper plate and the beams. Moreover, the fluid reaction is not taken into account. The simplification and the reductions of the number of integrals to be evaluated are inspired by Mace [START_REF] Mace | Periodically stiffened fluid-loaded plates, I: Responce to convected harmonic pressure and free wave propagation[END_REF][START_REF] Mace | Periodically stiffened fluid-loaded plates, II: Responce to line and point forces[END_REF]. The transformed displacement used in this system, discussed by Evseev [START_REF] Evseev | Sound radiaton from an infinite plate with periodic inhomogeneities[END_REF] and Mace [START_REF] Mace | Sound radiate from a plate reinforecd by two sets of parallel stiffeners[END_REF] and also in [START_REF] Brunskog | Prediction model for the impact sound level of lightweight floors[END_REF], can be written as w( ) = F R e i( x0+ y 0) S( ) ; ( ; en) 2 + 2 2 ; 4

(A4)

where, as before, e = 2 =land is the bending wave number. The fluid reaction is not included in equations (A2-A5) .

The inverse transform w(x y) = F ;1

x y f w( )g is defined in [START_REF] Brunskog | Prediction model for the impact sound level of lightweight floors[END_REF], equation (2 b). The half-way transform, transformed in the y; direction or inverse transformed in the x; direction, is denoted w(x ) = F ;1

x f w( )g = F y fw(x y)g.

Define a function for the sums in (A4-A5),

; ( ; en) 2 + 2 2

; 4 : (A6) Thus, equations (A4) and (A5) can be expressed in terms of , P a ( ) = F R e i y 0 ( x 0 )=D P b ( ) = ( 0 ) =D:

The function can be given explicitly using the Poisson's sum formula, contour integration and a geometric series expression, a method described in Mace [START_REF] Mace | Periodically stiffened fluid-loaded plates, I: Responce to convected harmonic pressure and free wave propagation[END_REF][START_REF] Mace | Periodically stiffened fluid-loaded plates, II: Responce to line and point forces[END_REF]. After considerable manipulations, equation (A6) can be expressed as ; 4 2 l ( x 0 ) = ie ;ix0 q; =q ;

1 ; e ;il( +q;) ; ie ix0q; =q ;

1 ; e ;il( ;q; ) + e ;x0 q+ =q + 1 ; e ;l(i +q+) ; e x0q+ =q + 1 ; e ;l(i ;q+) (A7) for 0 < x 0 < l, and where q + = p 2 + 2 and q ; = p 2 ; 2 . For other positions of x 0 , the periodicity of the structure can be used to translate the co-ordinates so that the inequality is fulfilled. However, the sum in (A6) converges rapidly (due to the fourth-order expression in the nominator) making it possible to use a truncated sum instead of the explicit expression (A7). When the fluid reaction is included, a truncated sum needs to be employed. The integrals in the inverse transform is now to be evaluated,

where the integrals I 1 and I 2 are defined. The second integral in (A8) is

In order to simplify the integration, subdivide the infinite integral I 2 into an infinite sum of finite integrals, A variable substitution 0 = ; 2n =l and a change of order between the sum and the integral, allow (A9) to be written as

The sum in (A11) is identified as 1 X n=;1 e ;i( +en)x S( 0 + en ) = ( ; 0 x)=D: Thus, I 2 becomes a finite integral,

Since the driving-point mobility is to be evaluated, x = x 0 . Utilising this and the fact that is symmetric in a if x = 0 , ( 0 ) = ( ; 0 ) , can be used to reduce the integral to

The first integral in (A8) is

After use of (A10), variable substitution and change of order between the sum and the integral and identifying , I 1 can be written as a finite integral. The negative side can be reduced using the symmetry of in , yielding

The integral in the -direction is now to be evaluated. The complete inverse transform is written as w(x 0 y 0 ) = 1 where the integrals J 1 and J 2 are defined. The second term is

The symmetry of the functions ( ; x 0 ) = ( x 0 ) and G(; ) = G( ) can be used to reduce the integral to

The first integral J 1 can be shown to be

It is better numerically, however, to evaluate this integral in the same way as J 2 . Thus, if the symmetry is used where (A20) is to be evaluated numerically. The drivingpoint mobility is found by using the results presented in sections 4.2 to 4.4.


Jin Yong Jeon
email: jyjeon@hanyang.ac.kr

Jong Kwan Ryu

Jeong Ho

Hideki Tachibana

Review of the Impact Ball in Evaluating Floor Impact Sound

The purpose of this study was to review the use of a new standard impactor, the impact ball, in evaluating heavy-weight impact sounds in multi-story reinforced concrete residential buildings. Physical properties such as impedance and force of the impact sources, including the impact ball and human activity, were measured. In particular, noise from the impact ball was analyzed and the relationships between sound levels and subjective responses were investigated. The results showed that the noise from the impact ball is similar to the noise of children running and jumping, and that subjective responses to the noise correlate well with Zwicker's Loudness model and the newly defined floor impact sound level (L i,F max,AW ). It was also found that the noise level of the impact ball is slightly higher than that of the bang machine, although the impact ball has a lower impact force. In addition, when the noise from the impact ball was evaluated under both laboratory and in-situ conditions, the allowable sound level was found to be 54 dB (L i,F max,AW ).

Introduction

In Korea, concern about the effects of floor impact sound has been increasing, along with the number of civil complaints. For the rational settlement of social disputes concerning floor impact noise, a method for objective evaluation of that noise is required. The most objective measure would be to use real impact sources such as jumping and running children, but such a measure is not reproducible. The next best method is to develop an impact source which accurately simulates real impact sounds and then evaluate subjective responses to the new impact source. This method should be simple and easy for end users to assess the performance of floor impact sound insulation.

The frequency characteristics of current standard impact sources such as the tapping machine and bang machine are different from those of real impact sources [START_REF] Jis A | Acoustics -measurement of floor impact sound insulation of buildings. Part 2: Method using standard heavy impact source[END_REF][START_REF] Shi | Sundbäck: An investigation of the characteristics of impact sound sources for impact sound insulation measurement[END_REF][START_REF] Tachibana | Development of new heavy and soft impact source for the assessment of floor impact sound insulation of buildings[END_REF]. Watters [START_REF] Watters | Impact-noise characteristics of female hardheeled foot traffic[END_REF] reported that the floor impact spectrum of lightweight impact noise (as the steel hammer of the tapping machine is lighter than any other impact sources) generated by a tapping machine is different from the spectrum of the hard-heeled foot traffic of women. In addition, Bodlund [START_REF] Bodlund | Alternative reference curves for evaluation of the impact sound insulation between dwellings[END_REF] found that on a concrete floor, the noise spectrum of a 75 kg male walking or running is dominated more by low frequency bands than is the noise spectrum produced by a tapping machine. Warnock [START_REF] Warnock | Low frequency impact sound transmission through floor systems[END_REF][START_REF] Warnock | Low frequency impact sound transmission through floor systems[END_REF] emphasized the use-fulness of the tapping machine, but also pointed out that it does not produce low frequency noise [START_REF] Warnock | Prospects for a test method for rating floor-toppings for use on joist floors[END_REF]. Furthermore, several publications have reported on the impedance levels of modified versions of the standard impact source for structure-borne noise [START_REF] Scholl | Impact sound insulation of timber floors: Interaction between source floor coverings and load bearing floor[END_REF][START_REF] Scholl | Impact sound insulation: The standard tapping machine shall learn to walk![END_REF][START_REF] Scholl | A modified tapping machine to represent a waling person[END_REF]. These efforts have led to the proposal of a modified standard tapping machine [START_REF]Acoustics -measurement of sound insulation in buildings and of building elements. Part 11: Laboratory measurements of the reduction of transmitted impact noise[END_REF]. Although this modified tapping machine reasonably simulates a human walking (300-500 N), it still cannot simulate the problematic impact noise such as children running (600-1,000 N) and jumping (2,000-3,000 N).

The low frequency floor impact sound pressure level generated by the modified tapping machine was lower than real running and jumping noise in reinforced concrete residential buildings. Although ISO 717-2: 1996(E) -Annex A suggests the use of additional weight to simulate the low frequency of light-weight impact sounds, there are difficulties applying the spectrum adaptation term 'C I '; namely, (a) there is a large difference in heavy-weight impact sound levels with different slabs, and (b) since the floor impact is an impulsive sound, the impact sound should be evaluated by L max not by L eq .

For the past thirty years, floor impact noise in woodframe and reinforced concrete (RC) houses has been evaluated with the bang machine in Japan and Korea. However, the impact force of the bang machine (4,200 N) is above the range of actual impact forces and may damage the structural components of wood-frame houses. Therefore, a new standard impactor with a lower impact force was needed. The impact ball was specifically developed to reduce the potential damage to structural components in wood-frame houses [START_REF] Yasuoka | Development of new heavy impact source part 1: Point of development and technique of experiments[END_REF]. As it is standardized for lightweight floor impact and more accurately approximates real impact noise, the impact ball has been adopted as the second standard impact source in Japan [START_REF] Jis A | Acoustics -measurement of floor impact sound insulation of buildings. Part 2: Method using standard heavy impact source[END_REF][START_REF] Tachibana | Development of new heavy and soft impact source for the assessment of floor impact sound insulation of buildings[END_REF][START_REF] Yasuoka | Development of new heavy impact source part 1: Point of development and technique of experiments[END_REF][START_REF] Tachibana | Development of a heavy and soft source for the assessment of floor impact sound insulation[END_REF][START_REF] Preis | Psychoacoustics studies on assessment of floor impact sounds[END_REF]. Tachibana et al. [START_REF] Tachibana | Development of new heavy and soft impact source for the assessment of floor impact sound insulation of buildings[END_REF] examined the actual performance of impact balls in different Japanese residential buildings (RC, wood and steel structures). When they measured the noise frequencies of four impact sources (tapping machine, bang machine, impact ball and jumping of a 25 kg child), they found that the characteristics of the impact ball were the most similar to the noise frequency characteristics of real impact noise. At present, the light-weight (tapping machine) and the two heavy-weight impact sources (bang machine and impact ball, so classified because of their peak levels) are used in Japan [START_REF] Jis A | Acoustics -measurement of floor impact sound insulation of buildings. Part 2: Method using standard heavy impact source[END_REF], but in Korea the impact ball has not yet been adopted as a standard impact source.

The structural system of multi-story residential buildings in Korea used in the present study is different from that of Japanese residential buildings. Apartment buildings in Korea typically contain a floor heating system which has a box frame-type RC structure with load-bearing walls and without columns and beams. These structures usually have hard floors and walls throughout the interior areas. Therefore, it is necessary to estimate the possible applications of an impact ball as a standard heavy-weight floor impact source in RC structures by comparing its characteristics with those of the bang machine and tapping machine. This study also compares the noise generated by the impact ball to real impact noises in a typical Korean RC residential structure. In addition, this study proposes categories of floor impact sound generated by the impact ball based on subjective evaluations.

Social survey

A survey of the major impact noise sources in 611 apartments in the Seoul area was conducted. The ages of children responsible for heavy-weight floor impact sound were recorded. As shown in Figure 1, the results indicate that children ages 6 to 9 years are the primary sources of floor impact sound. A recent survey in Korea on the national physique shows that the average weight of boys in this age range is 26.5 kg; that of girls is 25.6 kg.

The aggravating aspects of floor impact noise are the inconsistent spatial factors of noise generated by children [START_REF] Jeon | Subjective evaluation of floor impact noise based on the model of ACF/IACF[END_REF][START_REF] Jeon | Objective and subjective evaluation of floor impact noise[END_REF]. As shown in Figure 2, this study revealed that the jumping and running of children are the principal cause for the majority of complaints.

Characteristics of the impact ball

An impact ball (JIS A 1418-2) weighing 2.5 ±0.2 kg with a diameter of 185 mm was used. The thickness of the outer wall was 30 mm. The height of the free fall impact was 1 m with an impact time of 20 ms. Table I shows a comparison of the impact ball and the bang machine. In many respects the impact ball is more useful than the bang machine, including in its operation and maintenance.

Difference between the old and new impact ball

The old impact ball used in this study was developed just before a newly designed impact ball was released in 2001 by the Japanese company RION. Although the old one satisfies JIS A 1418-2, it is made of SBR (Styrene Butadiene Rubber) and its impact force varies with temperature [START_REF] Abe | Dependence on temperature of impact force characteristics in standard heavy impact sources. Part: 1 Equipment and techniques of experiments[END_REF][START_REF] Yasuoka | Dependence on temperature of impact force characteristics in standard heavy impact sources. Part: 2 Study on experimental results[END_REF][START_REF] Inoue | Dependence on temperature of impact force characteristics in standard heavy impact sources. Part: 3 Study on floor impact sound[END_REF]. The newer version is made with silicone rubber, which minimizes the effect of temperature. In addition, the effective mass, size and coefficient of restitution have been modified. The differences in the physical properties between the old and new balls are shown in Table II. We compared the frequency characteristics of the two impact balls in 1/3 octave bands as indicated in Figure 3. The impact balls were tested in a newly built multi-story apartment building. The floor structure consisted of a 150 mm concrete slab, heating pipes on 60 mm light-weight concrete (insulation) and 45 mm finishing mortar. The total area of an individual unit was 100 m 2 and the test was conducted in the living room and two bedrooms. As indicated in the 1/3 octave band analysis shown in Figure 3, the impact sound level of the new impact ball is, on average, approximately 1 dB lower than the that of the old impact ball at all frequencies except 63 Hz. The correlation coefficient for the spectrum of the two impact balls was 0.995 with a significance level of p <0.01.

Real versus standard impact sources

Impedance

In order to make a comparison of physical properties between standard impact sources and human impact sources, the mechanical impedance of each impact source was measured. The standard sources tested were the old impact ball, the tire of the bang machine, the original tapping machine and the modified tapping machine which is ISO 140-11 regulated. The human impact sources included an adult walking barefoot and a 7-year-old child.

For the impedance measurement, random excitation signals were sent to the impact sources through a shaker. Contrary to previous studies [START_REF] Scholl | Impact sound insulation of timber floors: Interaction between source floor coverings and load bearing floor[END_REF][START_REF] Scholl | Impact sound insulation: The standard tapping machine shall learn to walk![END_REF][START_REF] Scholl | A modified tapping machine to represent a waling person[END_REF], the shaker was placed horizontally in order to avoid the effect of the impact source's own mass, and the simulation impact source was bonded with epoxy to a 100 cm 2 aluminum plate. As shown in Figure 4, the aluminum plate, force transducer and accelerometer were installed, and the excitation force and vibration acceleration were measured simultaneously. The natural frequency of the plate was greater than 4 kHz and its impedance was excluded in the calculations of the impact sources' impedance. Figures 4a-c show the mea-surement setup for the impact ball, the tire of the bang machine and the steel hammer of the tapping machine. For the modified tapping machine, resilient material with similar properties as an elastic layer (dynamic stiffness = 34 MN/m 3 ± 10%, loss factor = 0.2) was placed under the hammers laid on the test floor. As shown in Fig- ure 4d, the measurement setup for human bare foot was similar, but the whole human body was hung in the air with a hammock to create the same measurement condition. Impedance of the adult (65 kg) and the 7-year-old (20 kg) child, both barefoot and with a straight leg, was measured.

As shown in Figure 5, the impedance of all impact sources increased as the frequency was increased, and decreased after the maximum value. The first resonance frequency of both the impact ball and the tire of the bang machine was about 20 Hz and that of the modified tapping machine was about 40 Hz. These values were much higher than the first resonance frequency of the human bare foot, which was about 4 Hz.

In comparison with previous studies [START_REF] Scholl | Impact sound insulation of timber floors: Interaction between source floor coverings and load bearing floor[END_REF][START_REF] Scholl | Impact sound insulation: The standard tapping machine shall learn to walk![END_REF][START_REF] Scholl | A modified tapping machine to represent a waling person[END_REF], these results were different at the low frequency band but similar at the high frequency band. However, the impedance characteristics of the human bare foot were almost the same as those found in human body vibration studies [START_REF] Kitazaki | A modal analysis of whole-body vertical vibration, using a finite element model of the hu-man body[END_REF]. Griffin [START_REF] Griffin | Handbook of human vibration[END_REF] reported that the impedance or apparent mass of the human body had a maximum value at 3-7 Hz and the frequency at the maximum value changed according to the magnitude of applied vibration and the subject's posture.

In the case of the tapping machine's steel hammer, resonance frequency was not observed because the steel hammer had rigid body characteristics. The spring stiffness of the modified tapping machine originated from its resilient material. The modified tapping machine had one degree of freedom in its system characteristics, attributed to the steel hammer mass.

As shown in Figure 5, the second, third and forth modes of impedance of the impact ball and tire were determined. For example, the second mode of the impact ball was 80 Hz but its impedance level was very small. Therefore, we can surmise that the floor impact vibration is mainly generated by the first resonance modes of the impact ball and tire. From these results, the impedance of each standard impact source was found to be somewhat different from human impact, but the resonance frequency and impedance level of the impact ball were most similar to those of a human bare foot.

In the experiment, the excitation force was transmitted through the plate to the fixed impact sources on the plate. These impact sources were fixed by various methods to specific areas rather than to points, and a change would occur in the fixed area or the contact area when impact sources hit the floor. The area of contact of the impact source with the floor was smaller than the projected surface area. In this case, the effect of the change in contact area on the first resonance mode was very small. This is because the stiffness of the first mode is mainly influenced by the stiffness of the impact source at the moment its shape is changed into the first mode. Therefore, impedance of the impact source will not change, even if the contact area continuously changes as the impact source affects the floor.

Impact force

As another physical property, the force from standard impact sources and human impact sources was measured. Figure 6 shows the impact force sensor (PF-10 & UA-06A, RION) used in this experiment. The four standard impact sources used in the impedance experiment were also tested.

As human impact sources, adults walking and a child (20 kg weight) jumping from a height of 30 cm and running were tested. The measured impact forces of the child's running and jumping were 600-800 N and 1,000-1,600 N, respectively. Figure 7 shows the results of the impact force measurements. As shown in Figure 7, the bang machine had the highest level of impact force at the 63 Hz band. The impact force spectrum of the impact ball was more similar to the impact of children than adults. The force spectrum of the modified tapping machine was similar to that of the adults; however, its impact force was very small compared to that of children.

Impact sound pressure level

The frequency characteristics of the noise generated by a 25 kg child jumping and running, the bang machine, the tapping machine and the old impact ball in an apartment building were compared. The floor impact sounds generated by the standard impactors were measured and recorded according to JIS A 1418 ("Measurement of floor impact sound insulation of buildings"). The floor was impacted at five positions (in the center and at four corners) in the living room on the upper floor. Measurements were recorded on the ceiling of the apartment directly under the impact positions.

Figure 8 compares the standard impactors with real impacts. Of the three standard impactors, the frequency characteristics of the impact ball were the most similar to those of a running child. The correlation coefficient between the sound levels of the impact ball and running child was 0.981 (p <0.01). However, the impact sound pressure levels of the impact ball were about 4 dB higher than those of a jumping child. The impact sound level of the bang machine was about 2-5 dB higher than the impact sound levels of a child jumping and running, respectively, but at low frequencies, the bang machine generated the highest sound pressure levels of any of the five impact sources. The tapping noise had much higher levels than any of the other impact sounds in the 250 Hz to 4 kHz bands. As shown in Figure 8, when evaluating floor impact sound, the impact ball had a higher level than the bang machine. The difference in frequency characteristics of the three impactors is due to the differences in their respective impact forces and impact times.

As shown in Table III, when the correlations between the real impact sound and the standard impact sounds were calculated, the real impact sound from the 25 kg child correlated well with that of the impact ball. The correlation between real impact sound and the tapping or bang machine noise was weaker. As shown in Figure 8, the difference in the A-weighted overall sound level between a child running and an impact ball is about 5 dB.

The impact sound levels at different drop heights

The impact ball was dropped from heights of between 5 cm and 120 cm (at 5 cm intervals from 5-20 cm; 10 cm intervals from 20-120 cm). The sound levels were measured in the master bedroom of the apartment. As shown in Figure 9, from 63 to 500 Hz in the standard octave band, the impact sound levels below the floor increase logarithmically with increases in drop height. The impact sound levels at 500 and 250 Hz were more sensitive to increases in drop height than those at lower frequencies. However, at the standard drop height of 100 cm, a drop height difference of about 10 cm was not significant (less than 1 dB).

Comparison of impact sound levels with other impactors

To determine the floor impact sound level, L i,F max,AW , inverse A-weighted impact sound pressure level in JIS [START_REF] Jis A | Rating of sound insulation in buildings and of building elements. Part 2: Floor impact sound insulation[END_REF] of a given specimen, and measured maximum impact sound levels (L max ) were plotted against four octave band frequencies from 63 to 500 Hz and compared to the reference inverse A-weighting contour (83, 73, 66, 60, 57 and 56 dB at the frequencies of 63, 125, 250, 500, 1000 and 2000 Hz, respectively).

The impact sound pressure level values from the tapping machine were calculated according to JIS A 1419-2 [START_REF] Jis A | Rating of sound insulation in buildings and of building elements. Part 2: Floor impact sound insulation[END_REF] (L ′ n,AW , the inverse A-weighted normalized impact sound pressure level for light-weight impact noise). JIS A 1419 uses a reference curve for impact sound levels (octave bands) which is different from the reference curve used in ISO 717-2. JIS A 1418 states that light-weight impact noise must have an equivalent sound pressure level (L eq ) in the frequency range of 125-2,000 Hz. The maximum deviation at any frequency should not exceed 10 dB; and the single value is read from the standard contour and the 500 Hz ordinate.

Although the direct comparison of two different types of levels such as L i,F max,AW of the bang machine and impact ball and L ′ n,AW of the tapping machine makes interpretation very difficult, it is necessary to compare the impact sound levels with different impactors used in a same floor. Correlations between the sound levels of the impact ball and the bang machine, and the impact ball and the tapping machine, were sought. The measurements were made on 32 concrete slabs of different sizes and thicknesses as shown in Figure 10. The R 2 value was low (0.24) between the impact ball and the tapping machine because the tapping noise is strongly affected by floor surface material. As shown in Figure 10, the R 2 values between the bang machine and the impact ball are relatively high (0.83) because both impacts are classified as heavy-weight impact sources due to their peak levels.

As shown in Figure 10, from the regression line, the level difference of the impact ball on the 32 different slabs was 20 dB (L i,F max,AW ), whereas that of the bang machine was 14 dB (L i,F max,AW ), and 12 dB (L ′ n,AW ) for the tapping machine. Among the three standard impact sources, the impact ball had the largest floor impact sound pressure level. From the result it also had the widest range in impact sound levels from the 32 box-frame type structures, thus making it more objective and useful in distinguishing the impact sound for isolated structures and materials than the other impact sources.

Subjective evaluation of floor impact sound 4.1. Floor impact sound measurements

Floor impact sound was measured in eight unoccupied units of a 28 storey building complex (RC, reinforcedconcrete structure) located in Seoul. The floor consisted of a 150 mm thick concrete slab, an 80 mm thick light-weight concrete insulation layer, heating pipes and a 50 mm layer of finished cement mortar. The total area of each unit was 132 m 2 and the test was conducted at 30 impact positions (6 rooms, 5 points per room) in the living room, kitchen and four bedrooms. The frequency characteristics of an adult jumping and running, an impact ball, a bang machine and a tapping machine were measured in the eight unfurnished apartments. When the upper floor was impacted at the center of the rooms, binaural sound signals were recorded through a dummy head for auditory evaluation. The impact and recording positions were the locations at which previous social survey respondents had experienced the most floor impacts. In this investigation, a 65 kg adult generated the jumping impact sound because arranging for a child to jump at night was difficult. Floor impact sound measurements and recordings were performed according to JIS A 1418. The floor structures consisted of one standard floor (P) and seven sound insulation structures. The sound insulation treatments are identified as follows: 'F' -a floating floor with a 10 mm impact isolator, 'W' -treatment of the walls with a 2 mm rubber sheet and a 9.5 mm gypsum board, and 'C' -treatment of the ceiling with a vibration hanger, a 2 mm rubber sheet and a 9.5 mm gypsum board. Thus 'FWC' means all three sound isolation treatments were applied to the box frame structure. The frequency characteristics of each of the five different impactors averaged from the eight units are shown in Figure 11. The figure illustrates that the noise of the impact ball is more similar to the adult jumping and running noises than to the child jumping noise (refer to Figure 8). The average L i,F max,AW difference of 5 dB between the adult and child jumping noises is due to the weight difference of 40 kg. The impact sound pressure level of the impact ball was higher than that of a child's jumping or running, and lower than that of an adult jumping or running. Therefore, the sound level of the impact ball can be taken as the actual, average impact sound level from tenants in reinforced concrete residential buildings.

Table IV shows the correlation coefficients between the frequency characteristics (L max ) of real jumping and running noises and those of the three standard impact sounds for each noise insulation construction. Of the three impact noise sources, the impact ball had the highest average correlation with the real noise. The bang machine had the highest correlation with the real jumping noise in the standard (P) and wall insulation (W) structures. The measured sound levels of the impact ball on treated structures are shown in Figure 12.

Auditory perception test

To select the best evaluation method for the impact ball, relationships between subjective responses and physical values such as 'Arithmetic Mean' (the mean value of octave band sound pressure level), L Amax , Inverse-A and Zwicker's loudness of the standard impact sources were analyzed.

Experiments were performed in a testing booth with approximately 25 dBA of background noise. An electrostatic headphone (STAX SRX-3030) was used for the binaural hearing. The noise source through the standard structure (P) was used as the standard stimulus, and 47 comparison stimuli were obtained from the treated (insulated) floor structures. Both standard and comparison stimuli were 1.6 seconds each and presented in random order. Subjective loudness was evaluated by pair on a five scale score. If the first stimulus was louder or much louder than the second stimulus, subjects marked +1 or +2, respectively, and marked 0 if the first and second stimuli were the same. If the second stimulus was louder or much louder than the first stimulus, subjects marked -1 or -2, respectively. Thirty subjects participated in the experiment. The subjects consisted of undergraduate and graduate students.

Table V shows the relationship between subjective responses and physical measurements. The R 2 value of L Amax with the impact ball was 0.70. The R 2 value of the Arithmetic Mean was 0.57. L Amax had a stronger correlation with the subjective response than the other indices.

The relationship between subjective response and Inverse-A (L i,F max,AW ) was 0.69.

Table V also shows the calculated relationship between Zwicker's loudness and the subjective responses, with an R 2 value of 0.74. The relationship between Zwicker's percentile loudness N 10 [START_REF] Zwicker | Psychoacoustics. Facts and models[END_REF] and subjective responses was 0.77. Therefore, the evaluation method best related to subjective responses is N 10 for the standard impactors. Tachibana [START_REF] Tachibana | Subjective assessment of indoor noises -basic experiments with artificial sound[END_REF] made the same conclusion in his subjective evaluation of major noise sources. Although the values of Zwicker's loudness for the impact ball are highly correlated with the subjective responses, Zwicker's psychoacoustic parameters are difficult to determine because of the equipment needed and the calculation methods used. Therefore, it is better to use measures such as L Amax or L i,F max,AW .

The relationship between the subjective evaluation and physical measurements of the tapping noise is also shown in Table V indicating that the standard tapping machine produces better correlation due to the steady-state aspect of the tapping machine noise. The R 2 value of L ′ n,w + C I,63-2000 was 0.73 and the R 2 value of the Arithmetic Mean was 0.84. The R 2 value for the relationship between subjective response and Inverse-A (L ′ n,AW ) was 0.79, for Zwicker's loudness, 0.84, and for N 10 , 0.88. Rindel and Rasmussen [START_REF] Rindel | Assessment of airborne and impact noise from neighbors[END_REF] indicated that L ′ n,w + C I,50-2500 calculated by ISO 717-2 (1996) [START_REF]Acoustics -Rating of sound insulation in buildings and of building elements -Part 2: Impact sound insulation[END_REF] is highly correlated with subjective responses for timber joist floors and for concrete floors with soft and hard floorings. However, unlike European floor structures, the arithmetic mean and the Inverse-A (L ′ n,AW ) values are more highly correlated with subjective responses in composite floor structures with floor heating systems.

Classes of floor impact sound using the impact ball

On-site auditory experiment

The purpose of the on-site experiment was to acquire a rating for the impact ball noise level according to the annoyance felt by subjects in a real life situation. Along with the impact ball, the tapping machine, bang machine, and an adult jumping were added as noise sources. Auditory experiments were conducted with 98 subjects in a living room of a multi-story residential building. Three questionnaire forms were used for the evaluation of the floor impact sound [START_REF] Jeon | Evaluation of floor impact sound insulation in reinforced concrete buildings[END_REF]. As shown in Table VI, the floor impact sound was evaluated according to its 'audibility' (Evaluation Scale 1), 'disturbance' (Evaluation Scale 2) and 'amenity' (Evaluation Scale 3). These three aspects of noise perception seem to affect people's annoyance with residential noises.

The values for the subjective evaluation were divided into three groups (1-3, 4-6, 7-9) to help reduce the subjects' difficulty in determining the points on the scale. In addition, after careful consideration of the groups' borders, the annoyance limit at which the noise became bothersome to people was intentionally set as Evaluation Point 4.

For on-site auditory experiments, the impact ball and tire were dropped from various heights. The tapping noise level was varied by changing floor materials. Subjects recorded their responses to the three questionnaire sheets while on the floor directly below the impact locations. The distribution and regression analyses of the floor impact sound levels (L i,F max,AW and L ′ n,W + C i ) corresponding to the average subjective responses were investigated.

The subjective evaluation resulted in a linear relationship between the subjects' responses and the floor impact sound level. The classes of floor impact sound levels according to subjective responses are shown in Table VI; the value of Evaluation Point 4 for the impact ball noise was 54 dB.

The relationship between the floor impact sound levels and the subjective evaluation scores are shown in Figure 13. The perceived floor impact sound level of a jumping adult is about 3 dB lower than the impact ball noise, and more than 10 dB lower than the bang machine noise.

The just noticeable differences (JNDs, minimum perceived noise levels) of the impact ball were investigated to evaluate the appropriate levels for floor impact sound classification. The auditory experiments were conducted using a pair comparison method of 10 postgraduate students in their late twenties. The sound pressure level of the stan- dard stimulus was increased by 1 dB increments, starting from 2 dB up to 5 dB. The subjects listened through headsets and judged the higher floor impact sound level within a test chamber. The duration of the sound source was 0.8 s with an interstimulus interval of 1 s. The stimuli were presented randomly to the subjects. It was found that the JNDs of the sound generated by the impact ball and tapping machine were about 2 dB. Eighty-six percent of the respondents recognized this 2 dB difference for the impact ball and eighty-nine percent recognized it in the tapping noise.

Criteria of floor impact sound

Although the limen of the impact ball noise was found to be about 3 dB among the respondents, subjective responses to the three different evaluation scales showed, on an average, a 5 dB level difference between evaluation classes. Rindel [START_REF] Rindel | Acoustic quality and sound insulation between dwellings[END_REF][START_REF] Rasmussen | Sound insulation of dwellings -Legal requirements in Europe and subjective evaluation of acoustical comfort[END_REF] also suggested 5 dB as the level difference for rating floor impact noise. In order to aid the tenants' understanding of the impact sound classes, an attempt was made to clarify the meaning of VDI 4100 classifications such as CAC (Classes of Acoustical Comfort) [START_REF]Schallschutz von Wohnungen[END_REF].

The meaning of 'annoyance limit' must be recognized as a starting point for annoyance and so used as a guideline for allowable limits. Table VII shows the proposed classes for floor impact sound levels generated with an impact ball. In this table, noise levels lower than 44 dB are classified as 'Class 1'. 'Class 2' noise levels are lower than 49 dB, and 'Class 3' noise levels are lower than 54 dB.

Discussion and conclusions

Of the three standard impact sources, the frequency characteristics of the impact ball noise were found to be the most similar to real impact sounds in a multi-story residential building. The impact ball noise was subjectively evaluated as being more similar to jumping noise than to the bang machine noise. Both the impact ball and jumping have higher noise levels in the 125 Hz band than the bang machine. The overall noise level of the adult jumping was slightly higher than that of the bang machine or the impact ball [START_REF] Inoue | Dependence on temperature of impact force characteristics in standard heavy impact sources. Part: 3 Study on floor impact sound[END_REF], whereas a jumping child produced noise that was 5 dB lower than the standard impactors.

The impact ball has the widest physical and subjective evaluation ranges among the three standard impactors, indicating that it should be the one used for evaluating the performance of reinforced concrete floor structures with different floor treatment. Correlation analyses between the subjective responses and physical measurements showed that the subjective responses to the impact ball correlated well with L i,F max,AW , whereas the impact ball noise had higher correlation with Zwicker's loudness.

Measurement results of the frequency characteristics for the three standard and the real impact sources in RC residential buildings demonstrated that noise from the impact ball was most similar to the noise of children jumping and running. Floor sounds generated with a bang machine have different characteristics because the impact force of a bang machine is much greater than that of a real source. The method for measuring floor impact sounds where the impact ball was the only source, and an evaluation method using inverse-A weighted floor impact sound pressure level (L i,F max,AW ), were both highly correlated with subjective responses.

The allowable sound pressure level of the impact ball, based on subjective evaluations in both the laboratory and in-situ conditions, was 54 dB (L i,F max,AW ). Three classes of impact ball noise were set up with 5 dB differences between grades. Still, the clarity and understandability of the impact ball noise questionnaire could be improved. This will be a part of a future work.

for the Next Generation of the Korea Ministry of Environment, and grant No. 10023489 from the Standardized Technology Development Project of Ministry of Commerce, Industry and Energy. We gratefully acknowledge the review and comments of Dr. A. C. C Warnock and W. Scholl. We also thank Prof. J. H. Park of Hanyang University for help on the measurement and analysis of impedance.

Figure 1 .

 1 Figure 1. Children's ages related to floor impact sound.

Figure 2 .

 2 Figure 2. Main sources of floor impact sound.

Figure 3 .

 3 Figure 3. Frequency characteristics of the old (JIS impact ball) and the new impact ball dropped on a concrete floor.

Figure 4 .

 4 Figure 4. Impedance measurement setup for each impact source. (a) Impact ball (b) Tire of bang machine (c) Steel hammer of tapping machine (d) Modified tapping machine.

Figure 5 .

 5 Figure 5. Measurement result of impedance for each impact source.

Figure 6 .

 6 Figure 6. Installation of impact force sensor (PF-10, RION).

Figure 7 .

 7 Figure 7. Impact force spectrum for standard impact sources and human impacts.

Figure 8 .

 8 Figure 8. Frequency characteristics of real impact sounds generated by a 26-kg child and by standard impactors.

Figure 9 .

 9 Figure 9. Sound levels of the impact ball dropped from different heights.

Figure 10 .

 10 Figure 10. Comparison of sound levels between the impact ball and other impactors.

Figure 11 .

 11 Figure 11. Frequency characteristics of impact sounds generated by a 65-kg adult and by standard impactors.

Figure 12 .

 12 Figure 12. Measured sound level of the impact ball on treated structures.

Table I .

 I Comparison of the impact ball with the bang machine.

		Impact Ball	Bang Machine
	Weight	2.5±0.1 kg	7.3±0.2 kg
	Handling	1 person	2 persons
	Preparation	Quick and easy	Complicated
	Electricity	Not required	Required
	Maintenance	None	Often
	Damage to structures	None	Wood structure

Table II .

 II JIS impact ball (old) vs. new impact ball.

		Old impact ball New impact ball
	Effective mass	2.5±0.2 kg	2.5±0.1 kg
	Restitution coefficient	0.7±0.1	0.8±0.1
	Diameter	185 mm	178 mm Shell
	thickness	30 mm	32 mm
	Material	SBR	Silicone rubber

Table III .

 III Correlation coefficients between real impact sources and standard impactors (p <0.01).

		Impact Ball	Tapping Mach.	Bang Mach.
	Jumping	0.97	0.71	0.92
	Running	0.98	0.69	0.95

Table IV .

 IV Correlation coefficients between real impact noise and standard sources (p <0.01). Abbreviations: IB: Impact ball, TM: Tapping machine, BM: Bang machine.

	(a) Jumping
	Insulated components	IB	TM	BM
	P (Plain)	0.97	0.65	0.97
	F (Floor)	0.94	0.49	0.88
	C (Ceiling)	0.94	0.36	0.94
	W (Wall)	0.97	0.52	0.98
	FC	0.97	0.70	0.95
	FW	0.98	0.62	0.94
	WC	0.98	0.57	0.93
	FWC	0.97	0.72	0.93
	Overall	0.97	0.58	0.94
	(b) Running
	Insulated components	IB	TM	BM
	P (Plain)	0.97	0.75	0.93
	F (Floor)	0.95	0.61	0.81
	C (Ceiling)	0.93	0.51	0.92
	W (Wall)	0.96	0.56	0.91
	FC	0.97	0.69	0.92
	FW	0.95	0.55	0.94
	WC	0.96	0.55	0.87
	FWC	0.95	0.76	0.90
	Overall	0.96	0.62	0.90

Table V .

 V Relationship between subjective responses and physical measurements. Abbreviations: IB: Impact ball, TM: Tapping machine.

		IB (R 2 )	TM (R 2 )
	Arithmetic mean	0.57	0.84
	L Amax	0.70	-
	Inverse-A	0.69	0.79
	L ′ n,W + C I,63-2000	-	0.73
	Loudness	0.74	0.84
	Percentile loudness N 10	0.77	0.88
	Figure 13. Relationship between floor impact sound and subjec-
	tive score.

Table VI .

 VI Impact sound pressure levels (L i,F max,AW ) vs. subjective scores.

	Annoyance	Subjective	Scale 1	Scale 2	Scale 3	L i,F max,AW
	Group	Score	Noisiness	Disturbance	Amenity	(dB)
		1	Hardly perceivable	At ease	Excellent	40
	Not annoying	2	Far-off noise	Not affected	Very fine	44
		3	Unconcerned	Undisturbed	Good	49
		4	Slightly heard	Detectable	Controllable	54
	Annoying	5	Heard	Noticeable	Endurable	59
		6	Clearly heard	Discernable	Yielding	64
		7	Noisy	Obvious	Unbearable	69
	Very annoying	8	Very noisy	Undoubted	Intolerable	74
		9	Extremely noisy	Serious	Let's move OUT!	79

Table VII .

 VII Classes of floor impact sound.

	Impact ball	Class 1	44 ≤ Class 2	49 ≤ Class 3
	(L i,F max,AW )	< 44 dB	< 49 dB	< 54 dB

Acknowledgement

This work was supported by grant No. 03-1-11-2-002 from the Core Environmental Technology Development Project