# The Effect of Uncertainty in the Excitation on the Vibration Input Power to a Structure

Authors: A. Putra; B. R. Mace
Journal: Advances in Acoustics and Vibration
 (2013)

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## Abstract

In structural dynamic systems, there is inevitable uncertainty in the input power from a source to a receiver. Apart from the nondeterministic properties of the source and receiver, there is also uncertainty in the excitation. This comes from the uncertainty of the forcing location on the receiver and, for multiple contact points, the relative phases, the force amplitude distribution at those points, and also their spatial separation. This paper investigates quantification of the uncertainty using possibilistic or probabilistic approaches. These provide the maximum and minimum bounds and the statistics of the input power, respectively. Expressions for the bounds, mean, and variance are presented. First the input power from multiple point forces acting on an infinite plate is examined. The problem is then extended to the input power to a finite plate described in terms of its modes. The uncertainty due to the force amplitude is also discussed. Finally, the contribution of moment excitation to the input power, which is often ignored in the calculation, is investigated. For all cases, frequency band-averaged results are presented.


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## Body

## 1. Introduction

The treatment of structure-borne sound sources remains a challenging problem. Structural excitation to a building floor, for example, by active components like pumps, compressors, fans, and motors, is an important mechanism of sound generation. To obtain an accurate prediction of the injected input power from such sources, both the source and the receiver must firstly be characterised. However in practical application, the variability of source and receiver properties including the lack of knowledge in the excitation force creates uncertainty in the input power. The problem is exacerbated because in practice there will usually be multiple contact points (typically four) and 6 degrees of freedom (3 for translation and 3 for rotation) at each, and that force and moment components at each contact point will contribute to the total input power. Therefore to assess the uncertainty, some quantification of the bounds, mean, and variance of the input power is of interest.The uncertainty in vibrational energy due to random properties, for example, dimensions, shapes, boundary conditions, and so on in a simple receiver structure, such as a plate, has been described by Langley and Brown [1, 2], where expressions for the mean vibrational energy and its variance were developed. A closed form solution was presented for the relative variance as a function of modal overlap factor and the nature of the excitation [1]. In [2], the analysis was extended to the ensemble average of the frequency band-averaged energy as a function of the frequency bandwidth. The analysis proceeded on the assumption that the natural frequencies form a random point process with statistics governed by, for example, the Gaussian orthogonal ensemble (GOE). The same type of analysis for a more complex system has also been proposed [3].Regardless of the receiving structure, the concept of source descriptor has been proposed to characterise a source based on its ability to deliver power without necessarily knowing any information about the receiver [4, 5]. Here, the concept of effective mobility [6, 7], that is, the ratio of the actual velocity at a point and in one direction, to the contributions of the excitations from all components and points, was used.The effective mobility concept was also employed in [8] to estimate the total power to an infinite beam through four contact points. The importance of having knowledge of the force distribution at the contact points was acknowledged. For this purpose, three simple force ratio assumptions were introduced, and the effects of force position, type of excitation, the structural loss factor, the receiver, and the number of contact points were investigated. It was found that the estimation is accurate only in the mass-controlled region and when the system approximated a symmetrical response. In a later paper [9], the force ratio was defined in terms of its statistical distribution. Simple expressions of the distribution at the mass-, stiffness-, and resonance-controlled regions were derived.Characterisation of a source in practice can be approached by using a reception plate method [10, 11] which is based on laboratory measurements, where from measured mobilities and surface mean-square velocities of the receiver, the free velocities and mobilities of the source can be extracted.This paper focuses only on the uncertainty in the excitation with the source and receiver assumed to be deterministic. The source may have multiple contact points. The uncertainty in input power due to the excitation phase, its location, and separation of the contact points is investigated. First some general comments are made. Broadband excitation is described, although only time-harmonic excitation is considered here with frequency averages subsequently being taken. The input power from multiple point forces to an infinite plate is examined to give an insight into the physical mechanisms involved. In practice, the receiver will have modes, although the modal overlap might be high. The input power to a finite plate is then analysed, where now the forcing location on the receiver becomes important. The mean and the variance of the input power averaged over force positions are investigated. The results are also presented in frequency-band averages.The uncertainty in the input power due to uncertainty in the force amplitude for multiple contact points is also discussed for a simple case of two contact points. Rather than dealing with force ratios between contact points, the sum of the squared magnitudes of the forces is assumed known.Finally, the inclusion of moments in the excitation is investigated and predictions of its contribution to the input power are made.


## 2. Input Power and Uncertainty Assessment

### 2.1. Time-Harmonic Excitation

Consider a vibrating source connected through a single orN contact points to a receiver. For a time-harmonic excitation at frequency ω, the input power is expressed as a function of mobility (or impedance) of the source and receiver [10, 12]. This requires knowledge of both source and receiver mobilities and the so-called blocked force or free velocity of the source. In general, the mobilities are matrices and the blocked forces or the free velocities are vectors, with the elements relating to the various translational and rotational degrees of freedoms (DOFs) at the contact points. In this paper, however, the analysis is made by assuming that the force excitation is known, and the source mobility is assumed to be much smaller than that of the receiver, as is usually the case in practice. The input power is therefore given by
(1)Pin=12Re{F~*Y~F~},
where F~=[F1ejϕ1F2ejϕ2⋯FNejϕN]T is the vector of the complex amplitudes of the time-harmonic forces and where * denotes the conjugate transpose. The ith force has a real magnitude Fi and phase ϕi. The mobilities of the receiver are represented by an N×N matrix Y~. In this section, only forces are considered. Moment excitation is discussed in Section 6.


### 2.2. Broadband Excitation

For a broadband excitation over a frequency bandB, the input power should be defined in terms of power spectral density and can be written as
(2)SPin=∑iN‍SiiRe{Y~ii}+∑i,ki≠kNRe{S~ik}Re{Y~ik},
where Sii and S~ik are the autospectral density and cross-spectral density of the forces, respectively, and Y~ik is the transfer mobility between points i and k. The cross-spectral densities would often in practice be difficult to measure. However, the coherence relates the auto- and cross-spectral densities of the excitation. Thus by assuming that only the autospectra of the forces are known, the cross-spectra are such that
(3)|S~ik|=±γ2SiiSkk,
where 0≤γ2≤1 is the coherence. This gives maximum and minimum bounds to the magnitude of the cross-spectral density. A rather similar approach is proposed by Evans and Moorhouse [13] for the case of a rigid body source, where the real part of the cross-spectral density is predicted using the available data of the autospectra and the calculated free velocities at the contact points from the source rigid body modes. Hence, this is limited only to the mass-controlled region of the source at very low frequencies. The comparison with measured data shows a good agreement. However, neither of these approaches are implemented in this paper. All forces are assumed to be time harmonic.


### 2.3. Uncertainty Quantification

Two approaches are employed to describe the uncertainty in the input power, namely, possibilistic and probabilistic approaches [14]. The possibilistic approach gives an interval description of the input power, which lies between lower and upper bounds; that is,
(4)Pin∈[Pin_Pin¯],
where Pin_ and Pin¯ are the minimum and maximum bounds and Pin is the interval variable. One example has been given in Section 2.2, where the input power can be bounded by using the spectral coherence data.The probabilistic approach gives information about the likelihood and probability of the input power. The variation is specified by a probability density functionΠ. If Π(z) is a continuous function of some variable z, the mean or the expected value of the input power and its variance are defined by
(5)μPin=E[Pin]=∫z‍Pin(z)Π(z)dz,σPin2=∫z‍Pin2(z)Π(z)dz-(μPin)2.


## 2.1. Time-Harmonic Excitation

Consider a vibrating source connected through a single orN contact points to a receiver. For a time-harmonic excitation at frequency ω, the input power is expressed as a function of mobility (or impedance) of the source and receiver [10, 12]. This requires knowledge of both source and receiver mobilities and the so-called blocked force or free velocity of the source. In general, the mobilities are matrices and the blocked forces or the free velocities are vectors, with the elements relating to the various translational and rotational degrees of freedoms (DOFs) at the contact points. In this paper, however, the analysis is made by assuming that the force excitation is known, and the source mobility is assumed to be much smaller than that of the receiver, as is usually the case in practice. The input power is therefore given by
(1)Pin=12Re{F~*Y~F~},
where F~=[F1ejϕ1F2ejϕ2⋯FNejϕN]T is the vector of the complex amplitudes of the time-harmonic forces and where * denotes the conjugate transpose. The ith force has a real magnitude Fi and phase ϕi. The mobilities of the receiver are represented by an N×N matrix Y~. In this section, only forces are considered. Moment excitation is discussed in Section 6.


## 2.2. Broadband Excitation

For a broadband excitation over a frequency bandB, the input power should be defined in terms of power spectral density and can be written as
(2)SPin=∑iN‍SiiRe{Y~ii}+∑i,ki≠kNRe{S~ik}Re{Y~ik},
where Sii and S~ik are the autospectral density and cross-spectral density of the forces, respectively, and Y~ik is the transfer mobility between points i and k. The cross-spectral densities would often in practice be difficult to measure. However, the coherence relates the auto- and cross-spectral densities of the excitation. Thus by assuming that only the autospectra of the forces are known, the cross-spectra are such that
(3)|S~ik|=±γ2SiiSkk,
where 0≤γ2≤1 is the coherence. This gives maximum and minimum bounds to the magnitude of the cross-spectral density. A rather similar approach is proposed by Evans and Moorhouse [13] for the case of a rigid body source, where the real part of the cross-spectral density is predicted using the available data of the autospectra and the calculated free velocities at the contact points from the source rigid body modes. Hence, this is limited only to the mass-controlled region of the source at very low frequencies. The comparison with measured data shows a good agreement. However, neither of these approaches are implemented in this paper. All forces are assumed to be time harmonic.


## 2.3. Uncertainty Quantification

Two approaches are employed to describe the uncertainty in the input power, namely, possibilistic and probabilistic approaches [14]. The possibilistic approach gives an interval description of the input power, which lies between lower and upper bounds; that is,
(4)Pin∈[Pin_Pin¯],
where Pin_ and Pin¯ are the minimum and maximum bounds and Pin is the interval variable. One example has been given in Section 2.2, where the input power can be bounded by using the spectral coherence data.The probabilistic approach gives information about the likelihood and probability of the input power. The variation is specified by a probability density functionΠ. If Π(z) is a continuous function of some variable z, the mean or the expected value of the input power and its variance are defined by
(5)μPin=E[Pin]=∫z‍Pin(z)Π(z)dz,σPin2=∫z‍Pin2(z)Π(z)dz-(μPin)2.


## 3. An Infinite Plate Receiver

The input power to an infinite plate, as an ideal simple structure, is first investigated. The point and transfer mobilities for an infinite plate are given by [15]
(6)Y~p=18Bem,Y~t=Y~p[H0(2)(kL)-2jπK0(kL)],
where H0(2) is the zeroth-order Hankel function of the second kind, K0 is the zeroth-order modified Bessel function of the second kind, m is the mass per unit area, and Be=Eh3/12(1-ν2) is the bending stiffness of the plate having Young’s modulus E, thickness h, and Poisson’s ratio ν. The input point mobility is purely real and independent of frequency, behaving as a damper. In the transfer mobility, the first and second terms represent propagating and near-field outgoing waves, respectively.Assume an infinite plate is excited by two point forces separated by a distanceL. From (1), the force vector is F={F1ejϕ1F2ejϕ2}T and the mobility is a 2×2 matrix. The total power is, thus, the sum of the input power at each location which yields
(7)Pin=12Re{Y~p}(F12+F22)+Re{Y~t}F1F2cosφ,
where φ=ϕ1-ϕ2 is the phase difference between the two forces. Note that Y~12=Y~21=Y~t. For the case of an infinite structure, the input mobility is the same at any position which implies that Y~11=Y~22=Y~p.


### 3.1. Dependence on the Contact Points Separation

When the structural wavelengthλ is much larger than the separation of the excitation points (λ≫L), the transfer mobility is approximately equal to the point mobility. Therefore, Y~p≈Y~t. For simplicity, if F1=F2=F, the input power becomes
(8)Pin=Re{Y~p}F2(1+cosφ).
From (8), the maximum and minimum input power are
(9)Pin¯=2F2Re{Y~p},forφ=2nπPin_=0,forφ=(2n+1)π,
where n is any integer.When the structural wavelength is much smaller thanL (λ≪L), the input power is
(10)Pin=(Re{Y~p}+Re{Y~t}cosφ)F2.
For small structural wavelength, the maximum and minimum input powers depend on Re{Yt}cosφ which indicates the dependency of the power on the phase and the distance between the excitation forces. Now since kL≫1, the real part of the transfer mobility in (6) can be expressed in its asymptotic form [16] as (see Appendix A)
(11)Re{Y~t}=Re{Y~p}2πkLcos(kL-π4).
The input power can thus be written as
(12)Pin=Re{Y~p}(1+2πkLcos(kL-π4)cosφ)F2.
The input power at higher frequencies is thus bounded by
(13)Pin_¯=(1±2πkL)Re{Y~p}F2.Figure1 shows the total input power when the two forces are in-phase (φ=0) and out-of-phase (φ=π) as a function of kL. Note that the input power has been normalised with respect to the input power from two point forces acting incoherently (equal to two times the power from a single point force). It can be seen that the power fluctuates around the value it would have if the two forces were applied independently. From (12), the power is minimum and maximum when kL=(2n+1)π for n=0,1,… for in-phase and out-of-phase forces, respectively. These are when, with respect to the wavelength, the two in-phase forces become out of phase and the out of phase forces become in phase. The intersection between the two curves at high kL is when φ=π/2. For kL<1, that is, when the forcing distance is less than half a structural wavelength, the total power is constant with frequency.Figure 1
The normalised input power to an infinite plate subjected to two in-phase (—) and out-of-phase (⋯) harmonic unit point forces (— grey line: max/min bounds at higher frequencies).


### 3.2. Random Phase

In practice, accurate information regarding the relative phase between the forces is generally not known. It might be assumed that all possible relative phases have equal probability. The probability density functionΠ of the phase is then constant and given by
(14)Π(φ)=12π,φ=[0,2π].
From (10), the mean power and its variance using (5) are
(15)μPin=Re{Y~p}F2,σPin2=12(Re{Y~t}F2)2.
It can be seen that the mean power equals that which would be given by two forces acting independently. The variance, hence, arises from interference between the two forces. As Y~t=Y~p for kL≪1, from (15), the normalised standard deviation is σ/μ=1/2. For kL≫1, σ decreases as kL increases (see (11)). Substituting (11) into (15) gives
(16)σ2μ2=(1πkL)cos2(kL-π4).
The bounds of the variance occur when cos2(kL-π/4)=1. The maximum normalised standard deviation is therefore σ¯/μ=1/πkL.When the separation distance of the contact points is uncertain, consequentlykL mod π becomes unknown, while 1/(kL) is more or less constant. Therefore by averaging (16) over all possible kL mod π, the variance can be expressed as
(17)〈σ2μ2〉kL=(1πkL)[1π∫-π/2π/2‍cos2(kL-π4)d(kL)]=12πkL.
Therefore, the normalised standard deviation due to uncertainty in kL is σ/μ=1/2πkL.Figure2 shows the input power for a number of possible relative phases of excitation. The mean power is a function of the point mobility which then gives a constant value with frequency. The normalised deviation of the power (1±σ/μ) lies between the maximum and minimum input power and bounded by 1±1/πkL. It can be seen that uncertainty in kL increases the deviation of the input power.Figure 2
The normalised input power to an infinite plate subjected to two harmonic unit point forces with various relative phases (grey lines):-- mean, -·- mean ± standard deviation, — (thick lines) max/min bounds, — mean ± bounds of standard deviation, and ⋯ mean ± bounds of standard deviation due to uncertainty in kL.


### 3.3. Four Contact Points

The case of four input excitations is more realistic in practice, for example, a vibrating machine with four feet. Figure3 shows an infinite plate excited by four harmonic point forces. In this case, from (1), the mobility is a 4×4 matrix. The diagonal elements are the point mobilities at each location, which for an infinite structure are equal to Y~p. The analysis is generally complicated. For simplicity, by assuming that the excitation positions form a rectangular shape, therefore, Y~12=Y~21=Y~34=Y~43; Y~13=Y~31=Y~24=Y~42 and Y~14=Y~41=Y~23=Y~32. The problem can be simplified numerically by choosing a force vector whose phases are relative to the phase of one reference force. The force vector can be rewritten as
(18)F~=[F1F2ejψ2F3ejψ3F4ejψ4]T,
where ψi is the relative phase of Fi with respect to F1. Thus, the total input power is given by
(19)Pin=12Re{Y~p}(F12+F22+F32+F42)+Re{Y~12}(F1F2cosψ2+F3F4cos(ψ3-ψ4))+Re{Y~13}(F1F3cosψ3+F2F4cos(ψ2-ψ4))+Re{Y~14}(F1F4cosψ4+F2F3cos(ψ2-ψ3)).
Note again in (19), that the input power is the sum of the powers that would be injected by the forces acting individually (the first term) and terms that depend on the relative phases of the forces (the remaining terms).Figure 3
An infinite plate excited by four harmonic point forces applied at the vertices of a rectangle.Thus, if equal probability of relative phase is assumed and the phases are uncorrelated, the probability density functionΠ can be expressed as
(20)Π(ψ2,ψ3,ψ4)=Π(ψ2)Π(ψ3)Π(ψ4),Π(ψi)=12π.
Using (5) and assuming all the forces have equal amplitudes, the mean power and the variance are given by
(21)μPin=2Re{Y~p}F2,(22)σPin2=12(Re{Y~12}F2)2+12(Re{Y~13}F2)2+12(Re{Y~14}F2)2+12(Re{Y~23}F2)2+12(Re{Y~24}F2)2+12(Re{Y~34}F2)2.
As in the case of two point forces in Section 3.2, the variance of the input power depends only on the transfer mobilities while the mean power is only a function of the point mobility. In general, the mean and the variance of input power averaged over all possible phases of excitation for N contact points for a rectangular separation can be written as
(23)μPin=12∑iN‍Re{Y~ii}Fi2,σPin2=12∑i=1N-1‍∑k=2i<kN(Re{Y~ik}FiFk)2.The bounds for the normalised input power are obtained by inserting the maximum transfer mobility from (11) into (19), which for equal force amplitudes yields
(24)Pin_¯μ=1±2πk(L1-1/2+L2-1/2+L3-1/2).
In the same way, substituting (11) into (22), the bounds for the normalised standard deviation are
(25)σ_¯μ=±12πk(L1-1+L2-1+L3-1)1/2,
where the positive and negative signs are for the maximum and minimum bounds, respectively.Following the same method for the case of two point forces, the bounds due to uncertainty inkL are given by
(26)σ_¯μ=±0.51πk(L1-1+L2-1+L3-1)1/2.Figure4 shows the input power with various relative phases of excitation. As in Figure 2, the power fluctuates at high kL around the mean value. The standard deviation shown is from (22). The first dip in the standard deviation corresponds to the frequency where the diagonal L3 equals half the structural wavelength, and the subsequent dips are related to L1 and L2. Equations (25) and (26) again show that the standard deviation reduces as the separation of the contact points increases.Figure 4
The normalised input power to an infinite plate subjected to four harmonic unit point forces with various phases (grey lines):-- mean, -·- mean ± standard deviation, — (thick lines) max/min bounds, — mean ± bounds of standard deviation, and ⋯ mean ± bounds of standard deviation due to uncertainty in kL.


## 3.1. Dependence on the Contact Points Separation

When the structural wavelengthλ is much larger than the separation of the excitation points (λ≫L), the transfer mobility is approximately equal to the point mobility. Therefore, Y~p≈Y~t. For simplicity, if F1=F2=F, the input power becomes
(8)Pin=Re{Y~p}F2(1+cosφ).
From (8), the maximum and minimum input power are
(9)Pin¯=2F2Re{Y~p},forφ=2nπPin_=0,forφ=(2n+1)π,
where n is any integer.When the structural wavelength is much smaller thanL (λ≪L), the input power is
(10)Pin=(Re{Y~p}+Re{Y~t}cosφ)F2.
For small structural wavelength, the maximum and minimum input powers depend on Re{Yt}cosφ which indicates the dependency of the power on the phase and the distance between the excitation forces. Now since kL≫1, the real part of the transfer mobility in (6) can be expressed in its asymptotic form [16] as (see Appendix A)
(11)Re{Y~t}=Re{Y~p}2πkLcos(kL-π4).
The input power can thus be written as
(12)Pin=Re{Y~p}(1+2πkLcos(kL-π4)cosφ)F2.
The input power at higher frequencies is thus bounded by
(13)Pin_¯=(1±2πkL)Re{Y~p}F2.Figure1 shows the total input power when the two forces are in-phase (φ=0) and out-of-phase (φ=π) as a function of kL. Note that the input power has been normalised with respect to the input power from two point forces acting incoherently (equal to two times the power from a single point force). It can be seen that the power fluctuates around the value it would have if the two forces were applied independently. From (12), the power is minimum and maximum when kL=(2n+1)π for n=0,1,… for in-phase and out-of-phase forces, respectively. These are when, with respect to the wavelength, the two in-phase forces become out of phase and the out of phase forces become in phase. The intersection between the two curves at high kL is when φ=π/2. For kL<1, that is, when the forcing distance is less than half a structural wavelength, the total power is constant with frequency.Figure 1
The normalised input power to an infinite plate subjected to two in-phase (—) and out-of-phase (⋯) harmonic unit point forces (— grey line: max/min bounds at higher frequencies).


## 3.2. Random Phase

In practice, accurate information regarding the relative phase between the forces is generally not known. It might be assumed that all possible relative phases have equal probability. The probability density functionΠ of the phase is then constant and given by
(14)Π(φ)=12π,φ=[0,2π].
From (10), the mean power and its variance using (5) are
(15)μPin=Re{Y~p}F2,σPin2=12(Re{Y~t}F2)2.
It can be seen that the mean power equals that which would be given by two forces acting independently. The variance, hence, arises from interference between the two forces. As Y~t=Y~p for kL≪1, from (15), the normalised standard deviation is σ/μ=1/2. For kL≫1, σ decreases as kL increases (see (11)). Substituting (11) into (15) gives
(16)σ2μ2=(1πkL)cos2(kL-π4).
The bounds of the variance occur when cos2(kL-π/4)=1. The maximum normalised standard deviation is therefore σ¯/μ=1/πkL.When the separation distance of the contact points is uncertain, consequentlykL mod π becomes unknown, while 1/(kL) is more or less constant. Therefore by averaging (16) over all possible kL mod π, the variance can be expressed as
(17)〈σ2μ2〉kL=(1πkL)[1π∫-π/2π/2‍cos2(kL-π4)d(kL)]=12πkL.
Therefore, the normalised standard deviation due to uncertainty in kL is σ/μ=1/2πkL.Figure2 shows the input power for a number of possible relative phases of excitation. The mean power is a function of the point mobility which then gives a constant value with frequency. The normalised deviation of the power (1±σ/μ) lies between the maximum and minimum input power and bounded by 1±1/πkL. It can be seen that uncertainty in kL increases the deviation of the input power.Figure 2
The normalised input power to an infinite plate subjected to two harmonic unit point forces with various relative phases (grey lines):-- mean, -·- mean ± standard deviation, — (thick lines) max/min bounds, — mean ± bounds of standard deviation, and ⋯ mean ± bounds of standard deviation due to uncertainty in kL.


## 3.3. Four Contact Points

The case of four input excitations is more realistic in practice, for example, a vibrating machine with four feet. Figure3 shows an infinite plate excited by four harmonic point forces. In this case, from (1), the mobility is a 4×4 matrix. The diagonal elements are the point mobilities at each location, which for an infinite structure are equal to Y~p. The analysis is generally complicated. For simplicity, by assuming that the excitation positions form a rectangular shape, therefore, Y~12=Y~21=Y~34=Y~43; Y~13=Y~31=Y~24=Y~42 and Y~14=Y~41=Y~23=Y~32. The problem can be simplified numerically by choosing a force vector whose phases are relative to the phase of one reference force. The force vector can be rewritten as
(18)F~=[F1F2ejψ2F3ejψ3F4ejψ4]T,
where ψi is the relative phase of Fi with respect to F1. Thus, the total input power is given by
(19)Pin=12Re{Y~p}(F12+F22+F32+F42)+Re{Y~12}(F1F2cosψ2+F3F4cos(ψ3-ψ4))+Re{Y~13}(F1F3cosψ3+F2F4cos(ψ2-ψ4))+Re{Y~14}(F1F4cosψ4+F2F3cos(ψ2-ψ3)).
Note again in (19), that the input power is the sum of the powers that would be injected by the forces acting individually (the first term) and terms that depend on the relative phases of the forces (the remaining terms).Figure 3
An infinite plate excited by four harmonic point forces applied at the vertices of a rectangle.Thus, if equal probability of relative phase is assumed and the phases are uncorrelated, the probability density functionΠ can be expressed as
(20)Π(ψ2,ψ3,ψ4)=Π(ψ2)Π(ψ3)Π(ψ4),Π(ψi)=12π.
Using (5) and assuming all the forces have equal amplitudes, the mean power and the variance are given by
(21)μPin=2Re{Y~p}F2,(22)σPin2=12(Re{Y~12}F2)2+12(Re{Y~13}F2)2+12(Re{Y~14}F2)2+12(Re{Y~23}F2)2+12(Re{Y~24}F2)2+12(Re{Y~34}F2)2.
As in the case of two point forces in Section 3.2, the variance of the input power depends only on the transfer mobilities while the mean power is only a function of the point mobility. In general, the mean and the variance of input power averaged over all possible phases of excitation for N contact points for a rectangular separation can be written as
(23)μPin=12∑iN‍Re{Y~ii}Fi2,σPin2=12∑i=1N-1‍∑k=2i<kN(Re{Y~ik}FiFk)2.The bounds for the normalised input power are obtained by inserting the maximum transfer mobility from (11) into (19), which for equal force amplitudes yields
(24)Pin_¯μ=1±2πk(L1-1/2+L2-1/2+L3-1/2).
In the same way, substituting (11) into (22), the bounds for the normalised standard deviation are
(25)σ_¯μ=±12πk(L1-1+L2-1+L3-1)1/2,
where the positive and negative signs are for the maximum and minimum bounds, respectively.Following the same method for the case of two point forces, the bounds due to uncertainty inkL are given by
(26)σ_¯μ=±0.51πk(L1-1+L2-1+L3-1)1/2.Figure4 shows the input power with various relative phases of excitation. As in Figure 2, the power fluctuates at high kL around the mean value. The standard deviation shown is from (22). The first dip in the standard deviation corresponds to the frequency where the diagonal L3 equals half the structural wavelength, and the subsequent dips are related to L1 and L2. Equations (25) and (26) again show that the standard deviation reduces as the separation of the contact points increases.Figure 4
The normalised input power to an infinite plate subjected to four harmonic unit point forces with various phases (grey lines):-- mean, -·- mean ± standard deviation, — (thick lines) max/min bounds, — mean ± bounds of standard deviation, and ⋯ mean ± bounds of standard deviation due to uncertainty in kL.


## 4. Input Power to a Finite Plate

For a finite structure, the mobility and, hence, the input power can be expressed in terms of a summation over modes of vibration. The mobility of a finite structure at an arbitrary point (x,y) subjected to a point force F at (x0,y0) at frequency ω is given by [10]
(27)Y~=jω∑n=1∞‍Φn(x0,y0)Φn(x,y)ωn2(1+jηn)-ω2,
where Φn is the nth mass-normalised mode shape of the structure, ωn is the nth natural frequency, and ηn is damping loss factor of the nth mode. A case often considered is that of a rectangular plate with simply supported boundary conditions as this system provides a simple analytical solution. For a simply supported rectangular plate with dimensions a×b, the mode shape and the natural frequency for mode (p,q) are
(28)Φpq(x,y)=2Msin(pπxa)sin(qπxb),ωpq=Bm[(pπa)2+(qπb)2],
where M is the total mass of the plate and p,q=1,2,3,…. The rigid body motion existing in the reception plate method [11] can be approximated by assuming the edges are guided or sliding. The mode shape functions involve cos terms instead of sin terms in (28). Using the corresponding real part of the point mobility in (27), the input power for point excitation is given by
(29)Pin=12∑p=1∞‍∑q=1∞‍ωωpq2ηnΦpq2(x0,y0)|F|2(ωpq2-ω2)2+(ωpq2ηn)2.


### 4.1. Averaging over Force Positions

Equation (29) implies that the input power depends on the forcing location which might be uncertain. Figure 5 shows the input power from a single point force for various forcing locations. The structure considered is an aluminium plate having dimensions 0.6×0.5×0.0015 m, Young’s modulus 7.1×1010 N/m2, ρ=2700 kg/m3, and damping loss factor η=0.05, assumed constant for every mode. The input power is normalised with respect to the input power to an infinite plate.Figure 5
The normalised input power to a finite plate subjected to a single harmonic point force for various possible force positions (grey lines): — mean and-·- mean ± standard deviation.The peaks occur at the resonances of the plate. They are distinct at low frequencies, but the modal overlap increases as the frequency increases. Analytically, since1/(ab)∫a‍∫b‍Φpq2dx0dy0=1/M, the mean input power averaged over all possible force positions is [10] as follows:
(30)〈Pin〉x0,y0=|F|22M∑p=1∞‍∑q=1∞‍ωωpq2η(ωpq2-ω2)2+(ωpq2η)2.
It can be seen that at high frequencies, the mean power converges to the same level as the input power to an infinite plate. The variance of the input power also decreases at high frequencies as with respect to the spatial variation of the vibration modes, the forcing location becomes less important as the frequency increases.


### 4.2. Averaging over Frequency Bands

Suppose that the excitation frequency lies between two frequenciesω1 and ω2. The input power can then be averaged over this frequency band and can be expressed as
(31)〈Pin〉ω=1ω2-ω1∫ω2ω1‍Pin(ω)dω.
Figure 6 shows the average input power for a 100 Hz bandwidth and at 50 Hz centre frequency spacing. For the plate example considered, the modal density is 0.065 modes/Hz. Therefore with this bandwidth, there are on average just over 6 modes in the band. It can be seen that the mean value is now close to the infinite plate value except below 100 Hz, where the modal overlap is low and the response is stiffness dominated.Figure 6
The normalised input power to a finite plate subjected to a single harmonic point force for various possible force positions averaged over 100 Hz frequency band (grey lines): — mean and-·- mean ± standard deviation.


### 4.3.  Prediction of Mean and Variance

The frequency average of the input power in (29) strongly depends on the statistical distribution of ω/((ωmn2-ω2)2+(ωmn2η)2). For a symmetric structure like a rectangular plate, asymptotically the natural frequency spacings have a Poisson distribution; that is, the spacings between successive natural frequencies are statistically independent and have an exponential distribution [17]. From [1, 2, 10, 17], the mean and variance of the input power averaged over all possible forcing locations yield
(32)〈Pin〉(ω,x0,y0)=|F|2ndπ4M,〈σ2〉(ω,x0,y0)=|F|4ndπ16ηωM2,
where nd(ω)=0.276ab/hcL is the modal density of plate and cL=(E/ρ(1-ν2))1/2 is the longitudinal plate wave speed. The mean power equals the input power to an infinite plate.Figure7 shows the normalised standard deviation of the input power averaged over force positions for different damping loss factors together with the modal overlap factor (MOF), where MOF=ndηω. It can be seen that there is a good agreement between the numerical calculations and the analytical predictions if MOF>1.The mean and standard deviation of the input power of a finite plate due to variation in forcing locations (numerical: — mean,-·- mean ± standard deviation, and (32): -- mean and ⋯ mean ± standard deviation; (a) η=0.05 and (b) η=0.1).
(a)
(b)From (32), the relative standard deviation can be defined as a function of modal overlap factor; that is,
(33)rσ=〈σ〉〈Pin〉=1πndηω=1πMOF.For plates with other shapes, the natural frequency spacing statistics are not Poisson (e.g., under many circumstances they asymptote to Gaussian orthogonal ensemble statistics). An alternative expression for the variance can then be found [1, 2].


### 4.4. Four Point Excitations

In this section, results are presented for the case where there are four rectangularly distributed point excitations. The diagram of the force positions is the same as in Figure3 for an infinite plate. The total input power is also similar to (19) except that for a finite structure the leading term is now expressed in terms of the input mobility for each force location. Therefore by also assuming equal force amplitudes and equal probability of all the excitation phases, the mean power for a finite plate subjected to four point forces according to (23) is given by
(34)μPin=12(Re{Y~11}+Re{Y~22}+Re{Y~33}+Re{Y~44})F2,
whilst the variance is the same as in (22).Figure8 shows the normalised mean and the standard deviation of the input power to a plate having dimensions 1×0.8×0.0015 m. The simulation is made for forces with a square distribution, where L1=L2=0.1 m (see Figure 3), and is located around the middle of the plate. The result shows typical behaviour where the variation is larger at low frequencies and decreases as kL increases.Figure 8
The normalised input power to a finite plate subjected to four harmonic point forces averaged over all possible excitation phases (— mean and-·- mean ± standard deviation).For the same force separation, Figure9 shows the relative standard deviation rσ for the average input power over all possible force positions and frequency bands. It can be seen that the standard deviation can be estimated reasonably and accurately by using results for the infinite plate, except at very low kL. The result from (33) is close to the averages of the numerical results. The figure also shows that the dips clearly correspond to frequencies where the distance between the excitation points is a multiple of half the structural wavelength, kL=nπ. However, Figure 9(b) shows that for a smaller damping, the infinite plate result is now an underestimate of the numerical values. In this case, the prediction using (33) gives a good agreement, particularly at large kL.The relative standard deviation of input power to a finite plate subjected to four harmonic point forces averaged over various force positions and frequency bands: — numerical,-·- infinite plate and -- (33) ((a) η=0.1 and (b) η=0.05).
(a)
(b)


## 4.1. Averaging over Force Positions

Equation (29) implies that the input power depends on the forcing location which might be uncertain. Figure 5 shows the input power from a single point force for various forcing locations. The structure considered is an aluminium plate having dimensions 0.6×0.5×0.0015 m, Young’s modulus 7.1×1010 N/m2, ρ=2700 kg/m3, and damping loss factor η=0.05, assumed constant for every mode. The input power is normalised with respect to the input power to an infinite plate.Figure 5
The normalised input power to a finite plate subjected to a single harmonic point force for various possible force positions (grey lines): — mean and-·- mean ± standard deviation.The peaks occur at the resonances of the plate. They are distinct at low frequencies, but the modal overlap increases as the frequency increases. Analytically, since1/(ab)∫a‍∫b‍Φpq2dx0dy0=1/M, the mean input power averaged over all possible force positions is [10] as follows:
(30)〈Pin〉x0,y0=|F|22M∑p=1∞‍∑q=1∞‍ωωpq2η(ωpq2-ω2)2+(ωpq2η)2.
It can be seen that at high frequencies, the mean power converges to the same level as the input power to an infinite plate. The variance of the input power also decreases at high frequencies as with respect to the spatial variation of the vibration modes, the forcing location becomes less important as the frequency increases.


## 4.2. Averaging over Frequency Bands

Suppose that the excitation frequency lies between two frequenciesω1 and ω2. The input power can then be averaged over this frequency band and can be expressed as
(31)〈Pin〉ω=1ω2-ω1∫ω2ω1‍Pin(ω)dω.
Figure 6 shows the average input power for a 100 Hz bandwidth and at 50 Hz centre frequency spacing. For the plate example considered, the modal density is 0.065 modes/Hz. Therefore with this bandwidth, there are on average just over 6 modes in the band. It can be seen that the mean value is now close to the infinite plate value except below 100 Hz, where the modal overlap is low and the response is stiffness dominated.Figure 6
The normalised input power to a finite plate subjected to a single harmonic point force for various possible force positions averaged over 100 Hz frequency band (grey lines): — mean and-·- mean ± standard deviation.


## 4.3.  Prediction of Mean and Variance

The frequency average of the input power in (29) strongly depends on the statistical distribution of ω/((ωmn2-ω2)2+(ωmn2η)2). For a symmetric structure like a rectangular plate, asymptotically the natural frequency spacings have a Poisson distribution; that is, the spacings between successive natural frequencies are statistically independent and have an exponential distribution [17]. From [1, 2, 10, 17], the mean and variance of the input power averaged over all possible forcing locations yield
(32)〈Pin〉(ω,x0,y0)=|F|2ndπ4M,〈σ2〉(ω,x0,y0)=|F|4ndπ16ηωM2,
where nd(ω)=0.276ab/hcL is the modal density of plate and cL=(E/ρ(1-ν2))1/2 is the longitudinal plate wave speed. The mean power equals the input power to an infinite plate.Figure7 shows the normalised standard deviation of the input power averaged over force positions for different damping loss factors together with the modal overlap factor (MOF), where MOF=ndηω. It can be seen that there is a good agreement between the numerical calculations and the analytical predictions if MOF>1.The mean and standard deviation of the input power of a finite plate due to variation in forcing locations (numerical: — mean,-·- mean ± standard deviation, and (32): -- mean and ⋯ mean ± standard deviation; (a) η=0.05 and (b) η=0.1).
(a)
(b)From (32), the relative standard deviation can be defined as a function of modal overlap factor; that is,
(33)rσ=〈σ〉〈Pin〉=1πndηω=1πMOF.For plates with other shapes, the natural frequency spacing statistics are not Poisson (e.g., under many circumstances they asymptote to Gaussian orthogonal ensemble statistics). An alternative expression for the variance can then be found [1, 2].


## 4.4. Four Point Excitations

In this section, results are presented for the case where there are four rectangularly distributed point excitations. The diagram of the force positions is the same as in Figure3 for an infinite plate. The total input power is also similar to (19) except that for a finite structure the leading term is now expressed in terms of the input mobility for each force location. Therefore by also assuming equal force amplitudes and equal probability of all the excitation phases, the mean power for a finite plate subjected to four point forces according to (23) is given by
(34)μPin=12(Re{Y~11}+Re{Y~22}+Re{Y~33}+Re{Y~44})F2,
whilst the variance is the same as in (22).Figure8 shows the normalised mean and the standard deviation of the input power to a plate having dimensions 1×0.8×0.0015 m. The simulation is made for forces with a square distribution, where L1=L2=0.1 m (see Figure 3), and is located around the middle of the plate. The result shows typical behaviour where the variation is larger at low frequencies and decreases as kL increases.Figure 8
The normalised input power to a finite plate subjected to four harmonic point forces averaged over all possible excitation phases (— mean and-·- mean ± standard deviation).For the same force separation, Figure9 shows the relative standard deviation rσ for the average input power over all possible force positions and frequency bands. It can be seen that the standard deviation can be estimated reasonably and accurately by using results for the infinite plate, except at very low kL. The result from (33) is close to the averages of the numerical results. The figure also shows that the dips clearly correspond to frequencies where the distance between the excitation points is a multiple of half the structural wavelength, kL=nπ. However, Figure 9(b) shows that for a smaller damping, the infinite plate result is now an underestimate of the numerical values. In this case, the prediction using (33) gives a good agreement, particularly at large kL.The relative standard deviation of input power to a finite plate subjected to four harmonic point forces averaged over various force positions and frequency bands: — numerical,-·- infinite plate and -- (33) ((a) η=0.1 and (b) η=0.05).
(a)
(b)


## 5. Random Force Amplitude

Besides the phases and locations of excitation, the force distribution at the contact points also generates variability in the input power. In practice, this could be due to the condition of the installation or the nature of the source itself. The problem is that it is very difficult in practice to have knowledge of its distribution in sufficient detail. The situation becomes more complicated still if moment excitation is also taken into consideration. Thus, in theoretical studies often, for convenience, only translational forces of equal amplitude are assumed [8, 11].Here, the approach is used to constrain the total square magnitude of the forces, that is, to assume(35)Feff2=∑i=1N‍Fi2
is constant. The goal is to obtain the maximum and minimum bounds of the mean and variance of the input power providing that only the so-called effective force Feff is known. From (23) for the mean input power to an infinite plate excited by two point forces,
(36)μPin=12Y~p(F12+F22)=12Y~pFeff2.
It can be seen that for an infinite plate, the mean power does not depend on the force distribution but rather on the characteristics of the receiver only. However, the variance depends on the product of the forces at the contact points. For two forces, the maximum variance is obtained when both contact points have equal force distribution; that is, F1=F2, and the minimum variance is when one of the forces equals the effective force while the other is zero; that is, F1=0; F2=Feff (see (23)). For a finite plate, the location of the excitation determines the mobility of the receiver. However, if the spatial variation is small (kL<π), the mean input power also depends only on the receiver mobility as in (36), where Y~p=Y~11≈Y~22.Figure10(a) shows the distribution of mean and the standard deviation of the input power due to random relative phases for various force amplitudes in frequency averages, where Feff2=1 N2. The excitation points are located at (0.25a,0.6b) and (0.58a,0.6b) for the same plate dimensions as in Section 4.1 with damping loss factor η=0.01. It can be seen that at low frequency, the variation is up to 4 dB which is quite significant. However, this reduces as the frequency increases. The variation can also be reduced by increasing the damping loss factors as shown in Figure 10(c).(a, c) The distribution of the mean (dark grey) and mean ± standard deviation (light grey) of the input power to a finite plate with two contact points due to random relative phases for various force amplitudes; (b, d) The average mean (—) and average mean ± standard deviation (-·-) over random force amplitudes. (a)-(b) η=0.01; (c)-(d) η=0.1.
(a)
(b)
(c)
(d)The average mean and variance from this results can be obtained by assumingF12 is uniformly probable between 0 and Feff2. Within this range, the probability density function Π can be assumed constant; that is,
(37)Π(F12)=1Feff2,F12=[0,Feff2].Using (5), the mean and the variance of the input power due to random phases averaged over random point force loading for the case of two contact points are given by
(38)〈μPin〉F12=14(Re{Y~11}+Re{Y~22})Feff2,〈σPin2〉F12=112(Re{Y~12}Feff2)2.Figures10(b) and 10(d) show the average mean and standard deviation of the input power from the results in Figures 10(a) and 10(c). Here, again it can be clearly seen that the deviation decreases with frequency and also as the damping loss factor increases. In this example, for η=0.1, the deviation of the power from the mean is insignificant above kL=5 (<1 dB).Figure11 shows the mean and variance due to random phases and force amplitudes for the infinite plate, also with two contact points, where in (38), Y~11=Y~22=Y~p. Asymptotic form of Y~t in (11) is used to calculate the maximum and minimum bounds (the case where F12=F22=Feff2/2). The range between the bounds can be seen to be roughly 2 dB between kL=4 and 16. However, this decreases as the frequency increases.Figure 11
The mean (--) and mean ± standard deviation (-·-) of the input power to an infinite plate with two contact points due to random relative phases and random force amplitudes (— max-min mean ± standard deviation, (thick line) max-min bounds).


## 6. The Contribution of Moment Excitation

### 6.1. The Effect of Moment Excitation on the Input Power

In the previous sections, only the translational force is considered as the driving excitation. However, in principle the motion at a contact point of a structure-borne source would involve up to six components where not only forces, but also moments will contribute to the total input power. The moment excitation is often neglected partly because of measurement difficulties rather than the fact that, in most cases, it gives a small contribution to the input power [18]. The contribution of moments is most important at higher frequencies and is less important than that of forces if the source is far away from discontinuities or boundaries [19].Figure12 illustrates the components of excitation assumed to act on a structure. The response at the contact point is a function of point mobilities, transfer mobilities for different axes, and also the cross-mobilities for different components. Therefore, there will be a 6×6 mobility matrix for each excitation point. The problem becomes more complicated for multiple contact points. For N contact points, the interaction between components will increase the size of the system matrix to 6N×6N.Figure 12
Six components of point excitations.In this paper, however, the problem is simplified by neglecting the in-plane excitations, that is,Fx, Fy, and Mz. Therefore, the mobility matrix is reduced to a 3×3 matrix for a single contact point. In general, the input power due to a combined point force and moment excitation can be rewritten as
(39)Pin=12(Re{Y~vF}|F~|2+Re{Y~vMM~F~*}+Re{Y~θ˙FF~M~*}+Re{Y~θ˙M}|M~|2),
where Y~vF and Y~θ˙M are the point force and point moment mobilities and Y~θ˙F and Y~vM are the cross-mobilities from force to rotation and from moment to translation, respectively. Since Y~θ˙F=Y~vM, hence Re{Y~vMM~F~*}+Re{Y~θ˙FF~M~*}=Re{Y~vM(M~F~*+F~M~*)}=2Re{Y~vM}Re{M~F~*}, and (39) can be rewritten as
(40)Pin=12(Re{Y~vF}|F~|2+2Re{Y~vM}Re{M~F~*}+Re{Y~θ˙M}|M~|2),
where, for an infinite plate, Re{Y~vM}=0.In matrix form, the power can be expressed as in (1) where F~=[F~M~xM~y]T is the vector of the force and moments. With inclusion of the moment excitation, the mobility matrix for a single contact point is given by
(41)Y~=[Y~vFY~vMxY~vMyY~θ˙xFY~θ˙xMxY~θ˙xMyY~θ˙yFY~θ˙yMxY~θ˙yMy],
where Y~ is symmetric.


### 6.2. Magnitude of Moment

The relative contribution to the input power depends of course on the magnitude of the excitation. Force and moment cannot be compared directly as they have different units. In a practical situation, they would also depend on the nature of the force generation mechanism in the source. The installation condition also has to be considered. The effects of moment excitation for a vibrating machine installed on soft support at the contact points would be different to those if the machine was bolted tightly to the receiver structure. Thus, the problem remains of qualifying the relative effects of force and moment. Moorhouse [18] proposed a dimensionless mobility where, for example, the real part of a cross-mobility, Re{Y~θ˙F}, is normalised by the real part of the corresponding point mobilities for both the force and the moment, Re{Y~θ˙M}Re{Y~vF}. This gives insight into the relative contribution due to the different excitation components.


#### 6.2.1. Single Point Excitation

The relative importance of force and moment in exciting a structure can be compared only in terms of their input power. However, to calculate the power not only the mobilities should be known but also the magnitudes and the phases of the excitation components (see (39)). Petersson [20] introduced the nondimensional eccentricity which relates the ratio of magnitude of moment and force to the structural wavenumber.Here, another approach is introduced where the magnitudes of the moments,M~=Mejϕ1, and the force, F~=Fejϕ2, at the contact point are related by an effective lever arm α by
(42)M=αF,
where 0<α<∞. This indicates that if α is very small, the structure is excited mainly by force, while if α is very large the structure is driven mainly by a moment. However, for convenience, a nondimensional unit is preferred to scale the relative input power. The total input powers, PF and PM, due to a force and a moment on an infinite plate are
(43)Pin=PF+PM=12Re{Y~vF}F2+12Re{Y~θ˙M}M2,
and the real parts of the point mobilities are given by
(44)Re{Y~vF}=ω8Bk2,Re{Y~θ˙M}=ω8B,
where all the cross-mobilities Re{Y~θ˙F} and Re{Y~vM} are zero. Consequently, the relative phase between the force and the moment is irrelevant.From (42), (43), and (44), the input power from moment excitation can be scaled in terms of the input power from the force by a nondimensional unit kα and is expressed as
(45)PM=(kα)2PF,
where k is the structural wavenumber. Equation (43) can be rewritten as
(46)Pin=((kα)2+1)PF.Figure13 shows the normalised total input power to an infinite plate for a single contact point. It can be seen that the power from force excitation is constant with frequency while the power from moment excitation is increasing with frequency. Both powers intersect at kα=1. For kα<1, the power is dominated by force excitation and for kα>1, the power is dominated by moment excitation.Figure 13
The normalised input power from force (-·-) and moment (--) excitations at a single contact point and the total power (—).For a finite plate receiver, the total input power is given as in (40). While the situation is now numerically complicated, (46) can again be used to scale the individual contribution to the input power.


#### 6.2.2. Multiple Point Excitation

Figure14 shows a diagram of a translational force F which generates moment M that can be resolved into moments Mx and My components. The moments can be expressed as
(47)Mx=LβFsin(δ),My=-LβFcos(δ),
where L is the lever arm, or the distance from the line of action of F to the point attached to the structure, δ is the angle between the lever arm and the positive x-axis, and β is a dimensionless scaling factor.Figure 14
The lever arm of the moment and force.Equations (42) and (47) can be used to define the relation between force and moment for multiple contact points. Figure 15 shows the forces and moments for a typical four point contact source, with the points having a rectangular distribution, where L32=L12+L22. The reference moment at any contact point might then be considered as a sum of contributions from forces at all the contact points. In this situation, the moment about the x-axis can thus be expressed in the form
(48){Mx,1Mx,2Mx,3Mx,4}=[α10-L2β3-L3β4sinθ0α2-L3β3sinθ-L2β4L2β1L3β2sinθα30L3β1sinθL2β20α4]×{F1F2F3F4},
and the moments about the y-axis are
(49){My,1My,2My,3My,4}=[α1L1β20L3β4cosθ-L1β1α2-L3β3cosθ00L3β2cosθα3L1β4-L3β1cosθ0-L1β3α4]×{F1F2F3F4}.
The subsequent sections discuss the effect of moment excitation on the input power to infinite and finite plates particularly for the multiple point excitation.Figure 15
The moment and force directions at source-receiver interface with four contact points.


### 6.3. Infinite Plate Receiver

#### 6.3.1. Single Contact Point

For a single contact point, Figure13 shows the input power as a function of kα and the relative phase between the force and moment is not important. However, for multiple contact points, the relative phases are required as the result of the coupling between forces and moments to the response at another contact point.


#### 6.3.2. Multiple Contact Points

As an example, for two contact points there are fifteen relative phases. Assume the distanceL between the two points is parallel with x-axis (δ=0) so that some transfer moment mobilities about x-axis become zero. The transfer moment mobilities are given in Appendix A. In this case, the total input power for the case of two contact points is given by
(50)Pin=12Re{Y~pvF}(F12+F22)+12Re{Y~pθ˙xMx}(Mx,12+Mx,22)+12Re{Y~pθ˙yMy}(My,12+My,22)+Re{Y~tvF}F1F2cosψ1+Re{Y~tθ˙xMx}Mx,1Mx,2cosψ2+Re{Y~tθ˙yMy}My,1My,2cosψ3+Re{Y~tvMy}(F2My,1cosψ4+F1My,2cosψ5),
where Yp denotes the point mobility (the same contact point) and Yt denotes the transfer mobility (different contact point). The phase ψ denotes the relative phase between the two components at the same or different contact points; for example, ψ4 is the relative phase between the moment about the y-axis and the force at different points. In (50), it has been noted that Y~tvMy=Y~tθ˙yF. Due to the complexity of this expression, it is difficult to determine the bounds of input power analytically. However for simplicity, it is assumed that all the components are in-phase, so that ψi=0 for i=1,2,3,4, and 5. By also assuming F1=F2=F, α1=α2 and following the same method as in (48) and (49) for a 2×2 matrix, thus, Mx,1=Mx,2=Mx=αF, My,1=(α+βL)F, and My,2=(α-βL)F. The asymptotic forms of the transfer mobility in (50) for this case can be expressed as (see also Appendix A) (51a)Y~tvMy=2kRe{Y~pθ˙yMy}×2πkL(sin(kL-π4)-jsin(kL-3π4)),(51b)Re{Y~tθ˙xMx}=2kLRe{Y~pθ˙xMx}2πkLsin(kL-π4),(51c)Re{Y~tθ˙yMy}=2Re{Y~pθ˙yMy}2πkL×(cos(kL-π4)-1kLsin(kL-π4)).By substituting (51a), (51b), and (51c) into (50) and setting the cos and sin terms equal to unity, the maximum and minimum bounds of the input power normalised with respect to the input power from translational force (PF) for in-phase excitation are found to be
(52)Pin¯_2PF=1+(kα)2+(kβL)22±2πkL[1+(kα)2-(kβL)2+(kβL)2kL].
This reduces to (13), the case where there is only translational force excitation, when kα≪1 and kβL≪1.In Section3.2, assuming random phases with equal probability in (50), the mean and the variance of the input power to an infinite plate receiver through N contact points can, in general, be expressed as
(53)μPin=12∑iN‍(Re{Y~iivF}Fi2+Re{Y~iiθ˙x,yMx,y}Mx,y(i)2),(54)σPin2=12∑i=1N-1‍∑k=2k>iN(Re{Y~ikvF}FiFk)2+12∑i=1N-1‍∑k=2k>iN(Re{Y~ikθ˙x,yMx,y}Mx,y(i)Mx,y(k))2+12∑i=1N-1‍∑k=2k>iN(Re{Y~ikθ˙x,yF}FiMx,y(k))2+12∑i=1N-1‍∑k=2k>iN(Re{Y~ikvMx,y}Mx,y(i)Fk)2,
where i and k indicate the ith and kth contact points, respectively.The bounds of the normalised standard deviation can be obtained by substituting (51a), (51b), and (51c) into (54). After algebraic manipulation, it can be approximated by
(55)σ¯_2PF≈1πkL[1+(kα)4(kL)2+((kα)2-(kβL)2)2+2((kα)2+(kβL)2)(kα)4(kL)2]1/2.
Again, for kα≪1 and kβL≪1, this yields the standard deviation for force excitation (see (16)). Following the same method in(17), the standard deviation due to uncertainty in the dimensionless spacing kL is given by
(56)σ¯_2PF≈12πkL[1+(kα)4(kL)2+((kα)2-(kβL)2)2+2((kα)2+(kβL)2)(kα)4(kL)2]1/2.Figure16 shows the mean input power and its standard deviation. The trend is the same as in Figure 2 for translational force excitation, except that the input power tends to increase at high frequencies due to moment excitations.Figure 16
The normalised input power to an infinite plate subjected to two harmonic unit point forces and two harmonic moments:-- mean, — (thick line) max/min bounds; (52), -·- mean ± standard deviation and — mean ± bounds of standard deviation; (55), ⋯ mean ± bounds of standard deviation due to uncertainty in kL and (56) (α=0.003 m, β=0.003).


### 6.4. Finite Plate Receiver

#### 6.4.1. Single Contact Point

For a finite plate, the phase difference between the force and moment becomes important as the cross-mobility is not zero. For the same plate dimensions as in Section4.1, Figure 17 shows the normalised input power against kα for a single contact point assuming in-phase force and moment. The result in Figure 17(a) shows the increase of the input power due to moment contribution at kα>0.25 (see also Figure 13). For the case where the excitation is near to the plate edge in Figure 17(b), the total power is significantly less at low kα, because the point mobility for force excitation (which dominates at low frequencies, kα≪1) is smaller near the edge. However, when kα>0.2, the input power is the same as that when the excitation position is near to the centre of the plate due to the increasing power from the moment, so that it compensates partly for the reducing power from the force.The normalised input power of a finite plate subjected to force and moment excitations at a single contact point ((a) the power with (--) and without (—) moment and (b) the total power for the contact point around the edge (--) and middle (—) of the plate: α=0.005 m, η=0.1).
(a)
(b)Figure18 shows the normalised input power for various forcing locations on the plate. The increase in the mean power due to the contribution of moment excitation can be seen roughly above kα>0.35.The normalised input power to a finite plate subjected to force and moment excitations at single contact point for various possible forcing locations (grey lines): — mean and-·- mean ± standard deviation. (a) discrete frequency and (b) frequency-band average; α=0.005 m, η=0.1.
(a)
(b)Figure19 shows the relative standard deviation rσ of the averaged input power for different damping loss factors and magnitudes of moment excitation. For all cases, it can be seen that the relative standard deviation, in an average sense, agrees reasonably well with that from the translational force from (33). This indicates that the ratio between the mean and standard deviation is approximately the same even if the moment excitation is neglected in the calculation of the input power. Particular attention is focused on the results at large kα when the moment starts to contribute substantially to the total input power.The relative standard deviation of the input power to a finite plate subjected to force and moment excitations at single contact point averaged over various possible forcing locations and frequency bands; — numerical and-- (33) ((a) α=0.005 m, η=0.1; (b) α=0.005 m, η=0.01; and (c) α=0.01 m, η=0.01).
(a)
(b)
(c)


#### 6.4.2.  Multiple Contact Points

Again, the relative phases due to coupling between forces and moments are of interest for multiple contact points. The mean power, assuming the relative phases between the excitations are equally probable, is the same as that in (53) for an infinite plate. However, for a finite plate a force will produce a rotation and a moment will produce a displacement at the same point. Therefore, the variance is given by
(57)σPin2=12∑i=1N-1‍∑k=2k>iN(Re{Y~vF(ik)}FiFk)2+12∑i=1N-1‍∑k=2k>iN(Re{Y~θ˙x,yMx,y(ik)}Mx,y(i)Mx,y(k))2+12∑i=1N‍∑k=1k≥iN(Re{Y~θ˙x,yF(ik)}FiMx,y(k))2+12∑i=1N‍∑k=1k≥iN(Re{Y~vMx,y(ik)}Mx,y(i)Fk)2.For four contact points, the mobility matrices are12×12. Using (53) and (57), Figure 20(a) shows the mean and standard deviation of the input power for damping loss factor η=0.05. The spatial separation of the contact points is again assumed to form a rectangular shape and L is the length of the diagonal. The results agree well with those from the infinite plate above kL=10. Below this, as in Section 4.4, the agreement deteriorates due to small damping. This is clearly shown in the relative standard deviation plotted in Figure 20(b). However, it can be seen that the numerical result has a good agreement with that from the prediction using (33).(a) The normalised input power to a finite plate subjected to force and moment excitations at four contact points averaged over various possible forcing locations and frequency bands: mean (— numerical calculation,-- infinite plate) and mean ± standard deviation (⋯ numerical calculation, -·- infinite plate). (b) The relative standard deviation of the input power; — numerical calculation, -·- infinite plate, and -- (33) (L=0.14 m, η=0.05, α=0.005 m, and β=0.005).
(a)
(b)Figure21 shows the relative standard deviation for η=0.01 for different distances between the contact points. Again, good agreement can be seen using (33), particularly for high frequencies. In Figure 21(b) differences are seen for kL>25, but they are less than 1 dB. At low kL, the prediction differs by 2 dB on average due to the very low-modal overlap. From the results presented, it shows that (33), which is applicable only for the translational force, can also be used to predict the contribution of moment at high frequencies.The relative standard deviation of the input power to a finite plate subjected to force and moment excitations at four contact points averaged over various possible forcing locations and frequency bands; — numerical calculation and-- (33) (η=0.01, α=0.005 m, and β=0.005; (a) L=0.07 m and (b) L=0.35 m).
(a)
(b)


## 6.1. The Effect of Moment Excitation on the Input Power

In the previous sections, only the translational force is considered as the driving excitation. However, in principle the motion at a contact point of a structure-borne source would involve up to six components where not only forces, but also moments will contribute to the total input power. The moment excitation is often neglected partly because of measurement difficulties rather than the fact that, in most cases, it gives a small contribution to the input power [18]. The contribution of moments is most important at higher frequencies and is less important than that of forces if the source is far away from discontinuities or boundaries [19].Figure12 illustrates the components of excitation assumed to act on a structure. The response at the contact point is a function of point mobilities, transfer mobilities for different axes, and also the cross-mobilities for different components. Therefore, there will be a 6×6 mobility matrix for each excitation point. The problem becomes more complicated for multiple contact points. For N contact points, the interaction between components will increase the size of the system matrix to 6N×6N.Figure 12
Six components of point excitations.In this paper, however, the problem is simplified by neglecting the in-plane excitations, that is,Fx, Fy, and Mz. Therefore, the mobility matrix is reduced to a 3×3 matrix for a single contact point. In general, the input power due to a combined point force and moment excitation can be rewritten as
(39)Pin=12(Re{Y~vF}|F~|2+Re{Y~vMM~F~*}+Re{Y~θ˙FF~M~*}+Re{Y~θ˙M}|M~|2),
where Y~vF and Y~θ˙M are the point force and point moment mobilities and Y~θ˙F and Y~vM are the cross-mobilities from force to rotation and from moment to translation, respectively. Since Y~θ˙F=Y~vM, hence Re{Y~vMM~F~*}+Re{Y~θ˙FF~M~*}=Re{Y~vM(M~F~*+F~M~*)}=2Re{Y~vM}Re{M~F~*}, and (39) can be rewritten as
(40)Pin=12(Re{Y~vF}|F~|2+2Re{Y~vM}Re{M~F~*}+Re{Y~θ˙M}|M~|2),
where, for an infinite plate, Re{Y~vM}=0.In matrix form, the power can be expressed as in (1) where F~=[F~M~xM~y]T is the vector of the force and moments. With inclusion of the moment excitation, the mobility matrix for a single contact point is given by
(41)Y~=[Y~vFY~vMxY~vMyY~θ˙xFY~θ˙xMxY~θ˙xMyY~θ˙yFY~θ˙yMxY~θ˙yMy],
where Y~ is symmetric.


## 6.2. Magnitude of Moment

The relative contribution to the input power depends of course on the magnitude of the excitation. Force and moment cannot be compared directly as they have different units. In a practical situation, they would also depend on the nature of the force generation mechanism in the source. The installation condition also has to be considered. The effects of moment excitation for a vibrating machine installed on soft support at the contact points would be different to those if the machine was bolted tightly to the receiver structure. Thus, the problem remains of qualifying the relative effects of force and moment. Moorhouse [18] proposed a dimensionless mobility where, for example, the real part of a cross-mobility, Re{Y~θ˙F}, is normalised by the real part of the corresponding point mobilities for both the force and the moment, Re{Y~θ˙M}Re{Y~vF}. This gives insight into the relative contribution due to the different excitation components.


### 6.2.1. Single Point Excitation

The relative importance of force and moment in exciting a structure can be compared only in terms of their input power. However, to calculate the power not only the mobilities should be known but also the magnitudes and the phases of the excitation components (see (39)). Petersson [20] introduced the nondimensional eccentricity which relates the ratio of magnitude of moment and force to the structural wavenumber.Here, another approach is introduced where the magnitudes of the moments,M~=Mejϕ1, and the force, F~=Fejϕ2, at the contact point are related by an effective lever arm α by
(42)M=αF,
where 0<α<∞. This indicates that if α is very small, the structure is excited mainly by force, while if α is very large the structure is driven mainly by a moment. However, for convenience, a nondimensional unit is preferred to scale the relative input power. The total input powers, PF and PM, due to a force and a moment on an infinite plate are
(43)Pin=PF+PM=12Re{Y~vF}F2+12Re{Y~θ˙M}M2,
and the real parts of the point mobilities are given by
(44)Re{Y~vF}=ω8Bk2,Re{Y~θ˙M}=ω8B,
where all the cross-mobilities Re{Y~θ˙F} and Re{Y~vM} are zero. Consequently, the relative phase between the force and the moment is irrelevant.From (42), (43), and (44), the input power from moment excitation can be scaled in terms of the input power from the force by a nondimensional unit kα and is expressed as
(45)PM=(kα)2PF,
where k is the structural wavenumber. Equation (43) can be rewritten as
(46)Pin=((kα)2+1)PF.Figure13 shows the normalised total input power to an infinite plate for a single contact point. It can be seen that the power from force excitation is constant with frequency while the power from moment excitation is increasing with frequency. Both powers intersect at kα=1. For kα<1, the power is dominated by force excitation and for kα>1, the power is dominated by moment excitation.Figure 13
The normalised input power from force (-·-) and moment (--) excitations at a single contact point and the total power (—).For a finite plate receiver, the total input power is given as in (40). While the situation is now numerically complicated, (46) can again be used to scale the individual contribution to the input power.


### 6.2.2. Multiple Point Excitation

Figure14 shows a diagram of a translational force F which generates moment M that can be resolved into moments Mx and My components. The moments can be expressed as
(47)Mx=LβFsin(δ),My=-LβFcos(δ),
where L is the lever arm, or the distance from the line of action of F to the point attached to the structure, δ is the angle between the lever arm and the positive x-axis, and β is a dimensionless scaling factor.Figure 14
The lever arm of the moment and force.Equations (42) and (47) can be used to define the relation between force and moment for multiple contact points. Figure 15 shows the forces and moments for a typical four point contact source, with the points having a rectangular distribution, where L32=L12+L22. The reference moment at any contact point might then be considered as a sum of contributions from forces at all the contact points. In this situation, the moment about the x-axis can thus be expressed in the form
(48){Mx,1Mx,2Mx,3Mx,4}=[α10-L2β3-L3β4sinθ0α2-L3β3sinθ-L2β4L2β1L3β2sinθα30L3β1sinθL2β20α4]×{F1F2F3F4},
and the moments about the y-axis are
(49){My,1My,2My,3My,4}=[α1L1β20L3β4cosθ-L1β1α2-L3β3cosθ00L3β2cosθα3L1β4-L3β1cosθ0-L1β3α4]×{F1F2F3F4}.
The subsequent sections discuss the effect of moment excitation on the input power to infinite and finite plates particularly for the multiple point excitation.Figure 15
The moment and force directions at source-receiver interface with four contact points.


## 6.2.1. Single Point Excitation

The relative importance of force and moment in exciting a structure can be compared only in terms of their input power. However, to calculate the power not only the mobilities should be known but also the magnitudes and the phases of the excitation components (see (39)). Petersson [20] introduced the nondimensional eccentricity which relates the ratio of magnitude of moment and force to the structural wavenumber.Here, another approach is introduced where the magnitudes of the moments,M~=Mejϕ1, and the force, F~=Fejϕ2, at the contact point are related by an effective lever arm α by
(42)M=αF,
where 0<α<∞. This indicates that if α is very small, the structure is excited mainly by force, while if α is very large the structure is driven mainly by a moment. However, for convenience, a nondimensional unit is preferred to scale the relative input power. The total input powers, PF and PM, due to a force and a moment on an infinite plate are
(43)Pin=PF+PM=12Re{Y~vF}F2+12Re{Y~θ˙M}M2,
and the real parts of the point mobilities are given by
(44)Re{Y~vF}=ω8Bk2,Re{Y~θ˙M}=ω8B,
where all the cross-mobilities Re{Y~θ˙F} and Re{Y~vM} are zero. Consequently, the relative phase between the force and the moment is irrelevant.From (42), (43), and (44), the input power from moment excitation can be scaled in terms of the input power from the force by a nondimensional unit kα and is expressed as
(45)PM=(kα)2PF,
where k is the structural wavenumber. Equation (43) can be rewritten as
(46)Pin=((kα)2+1)PF.Figure13 shows the normalised total input power to an infinite plate for a single contact point. It can be seen that the power from force excitation is constant with frequency while the power from moment excitation is increasing with frequency. Both powers intersect at kα=1. For kα<1, the power is dominated by force excitation and for kα>1, the power is dominated by moment excitation.Figure 13
The normalised input power from force (-·-) and moment (--) excitations at a single contact point and the total power (—).For a finite plate receiver, the total input power is given as in (40). While the situation is now numerically complicated, (46) can again be used to scale the individual contribution to the input power.


## 6.2.2. Multiple Point Excitation

Figure14 shows a diagram of a translational force F which generates moment M that can be resolved into moments Mx and My components. The moments can be expressed as
(47)Mx=LβFsin(δ),My=-LβFcos(δ),
where L is the lever arm, or the distance from the line of action of F to the point attached to the structure, δ is the angle between the lever arm and the positive x-axis, and β is a dimensionless scaling factor.Figure 14
The lever arm of the moment and force.Equations (42) and (47) can be used to define the relation between force and moment for multiple contact points. Figure 15 shows the forces and moments for a typical four point contact source, with the points having a rectangular distribution, where L32=L12+L22. The reference moment at any contact point might then be considered as a sum of contributions from forces at all the contact points. In this situation, the moment about the x-axis can thus be expressed in the form
(48){Mx,1Mx,2Mx,3Mx,4}=[α10-L2β3-L3β4sinθ0α2-L3β3sinθ-L2β4L2β1L3β2sinθα30L3β1sinθL2β20α4]×{F1F2F3F4},
and the moments about the y-axis are
(49){My,1My,2My,3My,4}=[α1L1β20L3β4cosθ-L1β1α2-L3β3cosθ00L3β2cosθα3L1β4-L3β1cosθ0-L1β3α4]×{F1F2F3F4}.
The subsequent sections discuss the effect of moment excitation on the input power to infinite and finite plates particularly for the multiple point excitation.Figure 15
The moment and force directions at source-receiver interface with four contact points.


## 6.3. Infinite Plate Receiver

### 6.3.1. Single Contact Point

For a single contact point, Figure13 shows the input power as a function of kα and the relative phase between the force and moment is not important. However, for multiple contact points, the relative phases are required as the result of the coupling between forces and moments to the response at another contact point.


### 6.3.2. Multiple Contact Points

As an example, for two contact points there are fifteen relative phases. Assume the distanceL between the two points is parallel with x-axis (δ=0) so that some transfer moment mobilities about x-axis become zero. The transfer moment mobilities are given in Appendix A. In this case, the total input power for the case of two contact points is given by
(50)Pin=12Re{Y~pvF}(F12+F22)+12Re{Y~pθ˙xMx}(Mx,12+Mx,22)+12Re{Y~pθ˙yMy}(My,12+My,22)+Re{Y~tvF}F1F2cosψ1+Re{Y~tθ˙xMx}Mx,1Mx,2cosψ2+Re{Y~tθ˙yMy}My,1My,2cosψ3+Re{Y~tvMy}(F2My,1cosψ4+F1My,2cosψ5),
where Yp denotes the point mobility (the same contact point) and Yt denotes the transfer mobility (different contact point). The phase ψ denotes the relative phase between the two components at the same or different contact points; for example, ψ4 is the relative phase between the moment about the y-axis and the force at different points. In (50), it has been noted that Y~tvMy=Y~tθ˙yF. Due to the complexity of this expression, it is difficult to determine the bounds of input power analytically. However for simplicity, it is assumed that all the components are in-phase, so that ψi=0 for i=1,2,3,4, and 5. By also assuming F1=F2=F, α1=α2 and following the same method as in (48) and (49) for a 2×2 matrix, thus, Mx,1=Mx,2=Mx=αF, My,1=(α+βL)F, and My,2=(α-βL)F. The asymptotic forms of the transfer mobility in (50) for this case can be expressed as (see also Appendix A) (51a)Y~tvMy=2kRe{Y~pθ˙yMy}×2πkL(sin(kL-π4)-jsin(kL-3π4)),(51b)Re{Y~tθ˙xMx}=2kLRe{Y~pθ˙xMx}2πkLsin(kL-π4),(51c)Re{Y~tθ˙yMy}=2Re{Y~pθ˙yMy}2πkL×(cos(kL-π4)-1kLsin(kL-π4)).By substituting (51a), (51b), and (51c) into (50) and setting the cos and sin terms equal to unity, the maximum and minimum bounds of the input power normalised with respect to the input power from translational force (PF) for in-phase excitation are found to be
(52)Pin¯_2PF=1+(kα)2+(kβL)22±2πkL[1+(kα)2-(kβL)2+(kβL)2kL].
This reduces to (13), the case where there is only translational force excitation, when kα≪1 and kβL≪1.In Section3.2, assuming random phases with equal probability in (50), the mean and the variance of the input power to an infinite plate receiver through N contact points can, in general, be expressed as
(53)μPin=12∑iN‍(Re{Y~iivF}Fi2+Re{Y~iiθ˙x,yMx,y}Mx,y(i)2),(54)σPin2=12∑i=1N-1‍∑k=2k>iN(Re{Y~ikvF}FiFk)2+12∑i=1N-1‍∑k=2k>iN(Re{Y~ikθ˙x,yMx,y}Mx,y(i)Mx,y(k))2+12∑i=1N-1‍∑k=2k>iN(Re{Y~ikθ˙x,yF}FiMx,y(k))2+12∑i=1N-1‍∑k=2k>iN(Re{Y~ikvMx,y}Mx,y(i)Fk)2,
where i and k indicate the ith and kth contact points, respectively.The bounds of the normalised standard deviation can be obtained by substituting (51a), (51b), and (51c) into (54). After algebraic manipulation, it can be approximated by
(55)σ¯_2PF≈1πkL[1+(kα)4(kL)2+((kα)2-(kβL)2)2+2((kα)2+(kβL)2)(kα)4(kL)2]1/2.
Again, for kα≪1 and kβL≪1, this yields the standard deviation for force excitation (see (16)). Following the same method in(17), the standard deviation due to uncertainty in the dimensionless spacing kL is given by
(56)σ¯_2PF≈12πkL[1+(kα)4(kL)2+((kα)2-(kβL)2)2+2((kα)2+(kβL)2)(kα)4(kL)2]1/2.Figure16 shows the mean input power and its standard deviation. The trend is the same as in Figure 2 for translational force excitation, except that the input power tends to increase at high frequencies due to moment excitations.Figure 16
The normalised input power to an infinite plate subjected to two harmonic unit point forces and two harmonic moments:-- mean, — (thick line) max/min bounds; (52), -·- mean ± standard deviation and — mean ± bounds of standard deviation; (55), ⋯ mean ± bounds of standard deviation due to uncertainty in kL and (56) (α=0.003 m, β=0.003).


## 6.3.1. Single Contact Point

For a single contact point, Figure13 shows the input power as a function of kα and the relative phase between the force and moment is not important. However, for multiple contact points, the relative phases are required as the result of the coupling between forces and moments to the response at another contact point.


## 6.3.2. Multiple Contact Points

As an example, for two contact points there are fifteen relative phases. Assume the distanceL between the two points is parallel with x-axis (δ=0) so that some transfer moment mobilities about x-axis become zero. The transfer moment mobilities are given in Appendix A. In this case, the total input power for the case of two contact points is given by
(50)Pin=12Re{Y~pvF}(F12+F22)+12Re{Y~pθ˙xMx}(Mx,12+Mx,22)+12Re{Y~pθ˙yMy}(My,12+My,22)+Re{Y~tvF}F1F2cosψ1+Re{Y~tθ˙xMx}Mx,1Mx,2cosψ2+Re{Y~tθ˙yMy}My,1My,2cosψ3+Re{Y~tvMy}(F2My,1cosψ4+F1My,2cosψ5),
where Yp denotes the point mobility (the same contact point) and Yt denotes the transfer mobility (different contact point). The phase ψ denotes the relative phase between the two components at the same or different contact points; for example, ψ4 is the relative phase between the moment about the y-axis and the force at different points. In (50), it has been noted that Y~tvMy=Y~tθ˙yF. Due to the complexity of this expression, it is difficult to determine the bounds of input power analytically. However for simplicity, it is assumed that all the components are in-phase, so that ψi=0 for i=1,2,3,4, and 5. By also assuming F1=F2=F, α1=α2 and following the same method as in (48) and (49) for a 2×2 matrix, thus, Mx,1=Mx,2=Mx=αF, My,1=(α+βL)F, and My,2=(α-βL)F. The asymptotic forms of the transfer mobility in (50) for this case can be expressed as (see also Appendix A) (51a)Y~tvMy=2kRe{Y~pθ˙yMy}×2πkL(sin(kL-π4)-jsin(kL-3π4)),(51b)Re{Y~tθ˙xMx}=2kLRe{Y~pθ˙xMx}2πkLsin(kL-π4),(51c)Re{Y~tθ˙yMy}=2Re{Y~pθ˙yMy}2πkL×(cos(kL-π4)-1kLsin(kL-π4)).By substituting (51a), (51b), and (51c) into (50) and setting the cos and sin terms equal to unity, the maximum and minimum bounds of the input power normalised with respect to the input power from translational force (PF) for in-phase excitation are found to be
(52)Pin¯_2PF=1+(kα)2+(kβL)22±2πkL[1+(kα)2-(kβL)2+(kβL)2kL].
This reduces to (13), the case where there is only translational force excitation, when kα≪1 and kβL≪1.In Section3.2, assuming random phases with equal probability in (50), the mean and the variance of the input power to an infinite plate receiver through N contact points can, in general, be expressed as
(53)μPin=12∑iN‍(Re{Y~iivF}Fi2+Re{Y~iiθ˙x,yMx,y}Mx,y(i)2),(54)σPin2=12∑i=1N-1‍∑k=2k>iN(Re{Y~ikvF}FiFk)2+12∑i=1N-1‍∑k=2k>iN(Re{Y~ikθ˙x,yMx,y}Mx,y(i)Mx,y(k))2+12∑i=1N-1‍∑k=2k>iN(Re{Y~ikθ˙x,yF}FiMx,y(k))2+12∑i=1N-1‍∑k=2k>iN(Re{Y~ikvMx,y}Mx,y(i)Fk)2,
where i and k indicate the ith and kth contact points, respectively.The bounds of the normalised standard deviation can be obtained by substituting (51a), (51b), and (51c) into (54). After algebraic manipulation, it can be approximated by
(55)σ¯_2PF≈1πkL[1+(kα)4(kL)2+((kα)2-(kβL)2)2+2((kα)2+(kβL)2)(kα)4(kL)2]1/2.
Again, for kα≪1 and kβL≪1, this yields the standard deviation for force excitation (see (16)). Following the same method in(17), the standard deviation due to uncertainty in the dimensionless spacing kL is given by
(56)σ¯_2PF≈12πkL[1+(kα)4(kL)2+((kα)2-(kβL)2)2+2((kα)2+(kβL)2)(kα)4(kL)2]1/2.Figure16 shows the mean input power and its standard deviation. The trend is the same as in Figure 2 for translational force excitation, except that the input power tends to increase at high frequencies due to moment excitations.Figure 16
The normalised input power to an infinite plate subjected to two harmonic unit point forces and two harmonic moments:-- mean, — (thick line) max/min bounds; (52), -·- mean ± standard deviation and — mean ± bounds of standard deviation; (55), ⋯ mean ± bounds of standard deviation due to uncertainty in kL and (56) (α=0.003 m, β=0.003).


## 6.4. Finite Plate Receiver

### 6.4.1. Single Contact Point

For a finite plate, the phase difference between the force and moment becomes important as the cross-mobility is not zero. For the same plate dimensions as in Section4.1, Figure 17 shows the normalised input power against kα for a single contact point assuming in-phase force and moment. The result in Figure 17(a) shows the increase of the input power due to moment contribution at kα>0.25 (see also Figure 13). For the case where the excitation is near to the plate edge in Figure 17(b), the total power is significantly less at low kα, because the point mobility for force excitation (which dominates at low frequencies, kα≪1) is smaller near the edge. However, when kα>0.2, the input power is the same as that when the excitation position is near to the centre of the plate due to the increasing power from the moment, so that it compensates partly for the reducing power from the force.The normalised input power of a finite plate subjected to force and moment excitations at a single contact point ((a) the power with (--) and without (—) moment and (b) the total power for the contact point around the edge (--) and middle (—) of the plate: α=0.005 m, η=0.1).
(a)
(b)Figure18 shows the normalised input power for various forcing locations on the plate. The increase in the mean power due to the contribution of moment excitation can be seen roughly above kα>0.35.The normalised input power to a finite plate subjected to force and moment excitations at single contact point for various possible forcing locations (grey lines): — mean and-·- mean ± standard deviation. (a) discrete frequency and (b) frequency-band average; α=0.005 m, η=0.1.
(a)
(b)Figure19 shows the relative standard deviation rσ of the averaged input power for different damping loss factors and magnitudes of moment excitation. For all cases, it can be seen that the relative standard deviation, in an average sense, agrees reasonably well with that from the translational force from (33). This indicates that the ratio between the mean and standard deviation is approximately the same even if the moment excitation is neglected in the calculation of the input power. Particular attention is focused on the results at large kα when the moment starts to contribute substantially to the total input power.The relative standard deviation of the input power to a finite plate subjected to force and moment excitations at single contact point averaged over various possible forcing locations and frequency bands; — numerical and-- (33) ((a) α=0.005 m, η=0.1; (b) α=0.005 m, η=0.01; and (c) α=0.01 m, η=0.01).
(a)
(b)
(c)


### 6.4.2.  Multiple Contact Points

Again, the relative phases due to coupling between forces and moments are of interest for multiple contact points. The mean power, assuming the relative phases between the excitations are equally probable, is the same as that in (53) for an infinite plate. However, for a finite plate a force will produce a rotation and a moment will produce a displacement at the same point. Therefore, the variance is given by
(57)σPin2=12∑i=1N-1‍∑k=2k>iN(Re{Y~vF(ik)}FiFk)2+12∑i=1N-1‍∑k=2k>iN(Re{Y~θ˙x,yMx,y(ik)}Mx,y(i)Mx,y(k))2+12∑i=1N‍∑k=1k≥iN(Re{Y~θ˙x,yF(ik)}FiMx,y(k))2+12∑i=1N‍∑k=1k≥iN(Re{Y~vMx,y(ik)}Mx,y(i)Fk)2.For four contact points, the mobility matrices are12×12. Using (53) and (57), Figure 20(a) shows the mean and standard deviation of the input power for damping loss factor η=0.05. The spatial separation of the contact points is again assumed to form a rectangular shape and L is the length of the diagonal. The results agree well with those from the infinite plate above kL=10. Below this, as in Section 4.4, the agreement deteriorates due to small damping. This is clearly shown in the relative standard deviation plotted in Figure 20(b). However, it can be seen that the numerical result has a good agreement with that from the prediction using (33).(a) The normalised input power to a finite plate subjected to force and moment excitations at four contact points averaged over various possible forcing locations and frequency bands: mean (— numerical calculation,-- infinite plate) and mean ± standard deviation (⋯ numerical calculation, -·- infinite plate). (b) The relative standard deviation of the input power; — numerical calculation, -·- infinite plate, and -- (33) (L=0.14 m, η=0.05, α=0.005 m, and β=0.005).
(a)
(b)Figure21 shows the relative standard deviation for η=0.01 for different distances between the contact points. Again, good agreement can be seen using (33), particularly for high frequencies. In Figure 21(b) differences are seen for kL>25, but they are less than 1 dB. At low kL, the prediction differs by 2 dB on average due to the very low-modal overlap. From the results presented, it shows that (33), which is applicable only for the translational force, can also be used to predict the contribution of moment at high frequencies.The relative standard deviation of the input power to a finite plate subjected to force and moment excitations at four contact points averaged over various possible forcing locations and frequency bands; — numerical calculation and-- (33) (η=0.01, α=0.005 m, and β=0.005; (a) L=0.07 m and (b) L=0.35 m).
(a)
(b)


## 6.4.1. Single Contact Point

For a finite plate, the phase difference between the force and moment becomes important as the cross-mobility is not zero. For the same plate dimensions as in Section4.1, Figure 17 shows the normalised input power against kα for a single contact point assuming in-phase force and moment. The result in Figure 17(a) shows the increase of the input power due to moment contribution at kα>0.25 (see also Figure 13). For the case where the excitation is near to the plate edge in Figure 17(b), the total power is significantly less at low kα, because the point mobility for force excitation (which dominates at low frequencies, kα≪1) is smaller near the edge. However, when kα>0.2, the input power is the same as that when the excitation position is near to the centre of the plate due to the increasing power from the moment, so that it compensates partly for the reducing power from the force.The normalised input power of a finite plate subjected to force and moment excitations at a single contact point ((a) the power with (--) and without (—) moment and (b) the total power for the contact point around the edge (--) and middle (—) of the plate: α=0.005 m, η=0.1).
(a)
(b)Figure18 shows the normalised input power for various forcing locations on the plate. The increase in the mean power due to the contribution of moment excitation can be seen roughly above kα>0.35.The normalised input power to a finite plate subjected to force and moment excitations at single contact point for various possible forcing locations (grey lines): — mean and-·- mean ± standard deviation. (a) discrete frequency and (b) frequency-band average; α=0.005 m, η=0.1.
(a)
(b)Figure19 shows the relative standard deviation rσ of the averaged input power for different damping loss factors and magnitudes of moment excitation. For all cases, it can be seen that the relative standard deviation, in an average sense, agrees reasonably well with that from the translational force from (33). This indicates that the ratio between the mean and standard deviation is approximately the same even if the moment excitation is neglected in the calculation of the input power. Particular attention is focused on the results at large kα when the moment starts to contribute substantially to the total input power.The relative standard deviation of the input power to a finite plate subjected to force and moment excitations at single contact point averaged over various possible forcing locations and frequency bands; — numerical and-- (33) ((a) α=0.005 m, η=0.1; (b) α=0.005 m, η=0.01; and (c) α=0.01 m, η=0.01).
(a)
(b)
(c)


## 6.4.2.  Multiple Contact Points

Again, the relative phases due to coupling between forces and moments are of interest for multiple contact points. The mean power, assuming the relative phases between the excitations are equally probable, is the same as that in (53) for an infinite plate. However, for a finite plate a force will produce a rotation and a moment will produce a displacement at the same point. Therefore, the variance is given by
(57)σPin2=12∑i=1N-1‍∑k=2k>iN(Re{Y~vF(ik)}FiFk)2+12∑i=1N-1‍∑k=2k>iN(Re{Y~θ˙x,yMx,y(ik)}Mx,y(i)Mx,y(k))2+12∑i=1N‍∑k=1k≥iN(Re{Y~θ˙x,yF(ik)}FiMx,y(k))2+12∑i=1N‍∑k=1k≥iN(Re{Y~vMx,y(ik)}Mx,y(i)Fk)2.For four contact points, the mobility matrices are12×12. Using (53) and (57), Figure 20(a) shows the mean and standard deviation of the input power for damping loss factor η=0.05. The spatial separation of the contact points is again assumed to form a rectangular shape and L is the length of the diagonal. The results agree well with those from the infinite plate above kL=10. Below this, as in Section 4.4, the agreement deteriorates due to small damping. This is clearly shown in the relative standard deviation plotted in Figure 20(b). However, it can be seen that the numerical result has a good agreement with that from the prediction using (33).(a) The normalised input power to a finite plate subjected to force and moment excitations at four contact points averaged over various possible forcing locations and frequency bands: mean (— numerical calculation,-- infinite plate) and mean ± standard deviation (⋯ numerical calculation, -·- infinite plate). (b) The relative standard deviation of the input power; — numerical calculation, -·- infinite plate, and -- (33) (L=0.14 m, η=0.05, α=0.005 m, and β=0.005).
(a)
(b)Figure21 shows the relative standard deviation for η=0.01 for different distances between the contact points. Again, good agreement can be seen using (33), particularly for high frequencies. In Figure 21(b) differences are seen for kL>25, but they are less than 1 dB. At low kL, the prediction differs by 2 dB on average due to the very low-modal overlap. From the results presented, it shows that (33), which is applicable only for the translational force, can also be used to predict the contribution of moment at high frequencies.The relative standard deviation of the input power to a finite plate subjected to force and moment excitations at four contact points averaged over various possible forcing locations and frequency bands; — numerical calculation and-- (33) (η=0.01, α=0.005 m, and β=0.005; (a) L=0.07 m and (b) L=0.35 m).
(a)
(b)


## 7. Conclusions

The uncertainty in input power to a structure due to uncertainty in the excitation has been investigated. For an infinite plate, the distance between the location where multiple forces are applied is not important if it is less than half a structural wavelength. The variance of the input power due to uncertainty in excitation phase and location tends to decrease as the nondimensional frequencykL increases. For multiple point excitation where the relative phases are random, the mean power and the variance depend only on the input and transfer mobility, respectively.As for the infinite plate, the variance of the input power to a finite plate also typically decreases as the frequency increases. The frequency average of the input power over all possible forcing locations from multiple contact points can be estimated reasonably and accurately by using the infinite plate result. However, for a very low damping (<5%) the agreement deteriorates and the simple prediction of the mean and variance can be used [1, 17].The uncertainty in the force amplitude at the contact points has also been discussed. Unless the spatial separation of the excitation locations is small, the distribution of the force amplitude through the contact points is important to obtain accurate estimates of the variation of the mean and standard deviation of the input power, particularly at low frequencies. This variation reduces as the damping loss factor is increased.The relative effect of moment excitation can be expressed in terms of a force and a distance corresponding to a characteristic of the source. It can also be scaled as a function of the input power of the force and the structural wavenumber. This effect tends to increase as frequency increases. The contribution to the total input power can be predicted using the simple expression of the relative standard deviation for the force. However in any event, the effects of moment excitation are typically small at low frequency and in any event are generally less than the effects of force excitation. They are typically, thus, of secondary importance.Finally, there remains the moot point of what uncertainty is, in practice, acceptable. This is to a large extent dependent on the typical uncertainty of machinery characterisation methods, such as the reception plate method. The attempt here is to quantify to some extent the uncertainty introduced by some details of the excitation, details that would typically not be measured.



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*Source: 478389-2013-08-27.xml*