Energy decay

If one considers the energy in a single subsystem, i, with power input only at tn=1

and then observes the exponential energy decay, the ratio of energies in consecutive time steps gives

E_i(t_n+1)

E_i(t_n) =e

−ωηΔt

tn ≠1 2.32

The energy in the single subsystem, which in this case is the source subsystem, will decay exponentially, by a factor e-ωηΔt_{, after the power input has ceased. Hence, an}

accurate solution is given when ωηΔt << 1. A suitably small time interval is proposed by Lyon and DeJong [18].

Δt ≤ 1

3ωη 2.33

Finite difference models often require the time interval to be as small as possible, with the limiting factor being computational efficiency [65], a smaller time interval means more calculations for a given duration.

Re-writing Eq. 2.33 and replacing the arbitrary factor in the denominator with the time interval constant, b, allows the time interval to be varied while still being inversely proportional to frequency and loss factor. It is proposed here that for practical purposes ηi should be the largest total loss factor in a frequency band for a

group of subsystems. As the total loss factor can vary with frequency (as well as angular frequency), hence the time interval can vary with frequency, giving a time interval for each frequency band.

Δt ≤ 1

bωη_i 2.34

There are other practical considerations that are taken into account when calculating the time interval. To calculate structure-borne transient power a fast Fourier transform is used (see section 3.6.1.2), this requires log2(fs) to equal an integer [66].

In practice the time interval calculated using Eq. 2.34 and then rounded down so that the sampling frequency, fs = 1/Δt, is equal to 2n, where n is an integer. This results in

a time interval that is a step-function with respect to frequency.

To assess whether the maximum level is affected by changes in the size of the time interval a single subsystem is used. The total loss factor for the subsystem chosen as the smallest total loss factor of materials used in TSEA models in the thesis, this is the total loss factor of aluminium, ηi = 0.001. The time interval constant is varied,

b = 3, 6, 12, 24, 48, 96. The fast-weighted maximum levels for each time interval constant are referenced to the fast-weighted maximum levels for b = 96, as the smallest time interval (largest sampling frequency) can be assumed to be numerically accurate [67].

63 125 250 500 1000 2000 4000 −1 −0.5 0 0.5 1

One−third octave−band centre frequency (Hz)

L F m ax, b − L F m ax, b = 96 (dB) b = 96 b = 3

Figure 2.2 Fast-weighted maximum level for a single subsystem, ηi = 0.001, for

varying time interval constants, all referenced to b = 96.

Figure 2.2 shows the worst case scenario where ηi = 0.001; however, it is seen that

errors in the fast-weighted maximum level do not exceed 1 dB across the frequency range. The magnitude of the error in the fast-weighted maximum levels is inversely proportional to the time interval constant, i.e. the errors increase as the size of the time interval increases. The ripple that is prominent in the fast-weighted maximum level for lower values of the time interval constant is caused by the step-function time-interval, as frequency increases and the step-function time interval forces the time interval to deviate from its original value the error in the fast weighted maximum level increases. This is a single subsystem with a very small loss factor and serves a purpose in investigating the relationship between the time interval constant and the maximum level, however it does not represent a group of coupled subsystems comprising of spaces and structures.

The same approach can be used for a group of subsystems with similar total loss factors. However plates and spaces often have very different loss factors. As energy returns to a subsystem from other subsystems, after the initial excitation, there may no longer be an exponential decay in the excited subsystem due to this returning energy. In order to investigate the error in predicting maximum levels caused by

varying the time interval constant, a seven subsystem model comprising of one room, and six plates is used. The walls and floors are 0.2 m thick concrete connected with L- junctions. All walls and floors radiate into the room, which has a reverberation time of 1.5 seconds for all frequency bands.

The same methodology is used as for the single subsystem example, the time interval constant is varied, b = 3, 6, 12, 24, 48, 96, when determining a frequency dependent time interval (Eq. 2.34). Power is input into a wall and the fast-weighted maximum level is evaluated for a number of subsystems.

63 125 250 500 1k 2k 4k −0.1 −0.05 0 0.05 0.1

One−third octave−band centre frequency (Hz) L v, F m ax, b − L v, F m ax, b = 96 (dB) b = 3 b = 96

Figure 2.3 Fast-weighted maximum velocity level for the source subsystem, Wall 4 (highlighted in red), for varying time interval constants, all referenced to b = 96.

63 125 250 500 1k 2k 4k −0.1 −0.05 0 0.05 0.1

One−third octave−band centre frequency (Hz) L p, F m ax, b − L p, F m ax, b = 96 (dB) b = 96 b = 3

Figure 2.4 Fast-weighted maximum sound pressure level for an adjacent subsystem, Room 1 (highlighted in blue), for varying time interval constants, all referenced to b = 96.

63 125 250 500 1k 2k 4k −0.1 −0.05 0 0.05 0.1

One−third octave−band centre frequency (Hz)

L v, F m ax, b − L v, F m ax, b = 96 (dB) b = 3 b = 96

Figure 2.5 Fast-weighted maximum velocity level for a non-adjacent subsystem, Wall 6 (highlighted in blue), for varying time interval constants, all referenced to b = 96.

For all three examples of fast-weighted maximum level in the seven subsystem idealised construction, Figures 2.3-2.5, the error due to varying the time interval constants does not exceed 0.05 dB. This error is lower for subsystems in the seven subsystem model than the single subsystem model, however the error does increase the further a subsystem is from the source subsystem but in this example it is not significant. By adding subsystems to the TSEA model the energy decay rate in each subsystem is not as large as for a single subsystem model, with no energy returning to from other subsystems. For TSEA models with more than one subsystem b = 6, is recommended as a balance between computational efficiency and accuracy.