Random Decrement Technique

The Random Decrement (RD) technique is a time domain system identification technique that generates the free-response of a linear single degree of freedom system from ambient excitation forces. The basic idea of the RD technique is to average time segments extracted from a response measurement, with the time segments being selected according to a trigger condition. The technique was initially developed to determine the dynamic characteristics and in-service damage detection of aerospace structures from measured responses [31]. The technique has the ability to estimate the amplitude dependence of the natural frequency and damping ratio for a mode of vibration.

The concept of the RD technique is explained by considering the following. For a system excited by random input forces, the response at time t₀ + t is composed of three parts: a step response of the structure from an initial displacement at t₀, an impulse response from an initial velocity at t0, and a random response due to the applied load in the time interval t. By selecting segments from the response time history that begin with a specified displacement response amplitude, and then combining these segments via superposition, it is observed that as the number of segments increases, the random response part will tend to zero. Furthermore, the sign of the initial velocity is expected to vary randomly with time, resulting in a zero initial velocity. Since the impulse response and random response parts are zero, the remaining part is the step response from the initial displacement. The resulting curve from this process is known as the random decrement signature.

The concept of the RD technique is described in Figure 4.9. Mathematical proofs

Figure 4.9: Random decrement technique (Source: Cole 1973 [31])

of the RD technique have been conducted [144, 177] to establish generalised models of the random decrement signature. The generalised models found the RD function to be proportional to the auto correlation function of an ergodic Gaussian distribu-tion [177]. This propordistribu-tionality between the RD funcdistribu-tion and the auto correladistribu-tion function means the RD technique is directly comparable with spectral density func-tions, since the correlation functions are attained from the inverse Fourier transform of spectral density functions.

For a lightly damped structure excited by random input forces, such as wind loading, the step response represents the single degree of freedom free vibration trace that is produced when the structure is initially displaced by the distance used in the segment selection process described above, and allowed to decay back to a zero amplitude. The natural frequency and damping ratio of the system can be extracted from the decay trace using numerous techniques, such as the logarithmic decrement or curve fitting discussed in Section 4.3.3. The step response derived from an initial displacement has been used in the description of the RD technique, but any response type — displacement, velocity, acceleration — can be used with the technique.

An important aspect in the application of the RD technique is the trigger con-dition used to select the segments for averaging. Consider the RD signature defined by;

D_xx(τ ) = E[x(t + τ )|x(t) = a^r] (4.72)

where D_xx(τ ) is an estimate of the RD function over the time lag τ , and E[x|T^c] is the conditional expected value of x for the condition defined by T_c. Equation (4.72) represents the RD function estimate for a level crossing trigger condition, with the level being defined by the reference amplitude a_r. Assuming the response x(t) is ergodic, the expected value is replaced with the time average to produce;

Dxx(τ ) = 1

where N_s is the number of segments used to generate the RD signature.

Another trigger condition is the use of peak amplitudes in the response time history, and is known as the Ranked Random Decrement [163].

D_xx(τ ) = 1

Two conditions are specified in the ranked RD technique: ˙x(t_i) = 0 requires x(t) to be a peak when the response is at the reference amplitude specified by x(t_i) = a_r. Since practical applications of the RD technique use discrete time sampled response data, the probability of x(t_i) = a_r occurring in the time series is low without careful selection of the reference amplitude. Depending on the amount of response data available, this can lead to insufficient segments being selected, which can cause significant errors in the estimation of the dynamic characteristics.

Therefore, a reference amplitude tolerance ∆a_r is introduced to broaden the extents of the acceptance interval about the reference amplitude, and thereby increase the probability of trigger events occurring. The estimate of the RD function is given by;

D_xx(τ ) = 1 The ranked RD technique is described in Figure 4.10. The trigger condition defined by the ranked RD technique can be used to estimate the dynamic characteristics for a particular response amplitude. By generating multiple RD functions for incre-mental values of the reference amplitude, the amplitude dependence of the dynamic characteristics can be observed.

In addition to the selection of a suitable trigger condition, the practical imple-mentation of the RD technique requires consideration for the following aspects:

• The response data used to generate the RD function must be for a single degree of freedom system. All structures have multiple modes of vibration, which means the vibration mode of interest must be isolated prior to applying the RD technique. This can be achieved with bandpass filtering the response data.

D_xx(T_c1;

T_c1

Dxx(Tc2;

t T_c2

Figure 4.10: Ranked RD concept (Source: Tamura and Suganuma 1996 [163] with notation adjustments)

Careful selection of the upper and lower cut-off frequencies is required to ensure unwanted modes are removed, without altering the response information for the mode of interest.

• A suitable reference amplitude tolerance for the ranked RD technique is re-quired to ensure sufficient segments are used to generate the RD function. As the amount of response data increases, the reference amplitude can decrease, because the reduced probability of detecting a peak in the narrower trigger range is offset by the increased quantity of response data, thereby maintaining the number of segments used. For a reduced quantity of response data, the reference amplitude tolerance must increase the number of segments detected, whilst not influencing the estimates of natural frequency and damping ratio.

This is particularly important when using the RD function for investigating the amplitude dependence of the dynamic characteristics.

• Establishing a minimum threshold for the number of segments (N^s) used to generate the RD function is important for maintaining the accuracy of the natural frequencies and damping ratios. The estimation of the damping ratio is expected to be more sensitive to the number of segments used. Previous research has recommended the minimum number of segments to be 2000 [27, 163], although stability in the damping ratio estimates has been observed for as little as 200 segments [22, 90].

• The number of cycles (N^c) used to extract the dynamic characteristics from

the RD function is particularly important when investigating the amplitude dependence of the dynamic characteristics. The variance of the RD function increases with the number of cycles, which implies the first cycle is expected to provide a more accurate damping ratio estimate. However, the damping estimate errors using the first cycle is found to be high, and it has been rec-ommended that improved results are obtained using five cycles [90]. This recommendation is in contrast to other research that has observed increasing overestimation of the damping ratio as the number of cycles increases [163].

Furthermore, there is the potential to dilute the amplitude dependent effects on the damping ratio estimates by increasing the number of cycles. This will introduce the latter parts of the RD function, which are not associated with the response amplitude of interest, to the estimation process.

As for the FDD and SSI techniques, the RD technique requires the measured re-sponse to be from a system excited by a Gaussian, white noise input force with zero mean. Furthermore, the system is assumed to have a single degree of freedom with linear, or small non-linear behaviour [73]. compared with FDD and SSI techniques, the RD technique is less complex to implement since only three relatively simple steps are performed: detection of trigger points, extraction of the time history seg-ments, and averaging of the time history segments. However, the RD technique requires large quantities of data in order to achieve satisfactory estimates of the natural frequency and damping ratio. Therefore, the RD technique is applied to the data obtained from the wind-induced monitoring programme described in Section 3.5, in order to determine the amplitude dependence of the natural frequencies and damping ratios of Latitude tower.

Example: Random Decrement Technique

The Random Decrement analysis used in this research was programmed using Mat-lab R2007b [170]. Figure 4.11 displays the RD function generated from multiple re-sponse time histories collected during the monitoring programme at Latitude tower.

The ranked RD technique was used with the following trigger condition parameters:

a_r = 0.3 mg_n, ∆a_r = 0.05. A tenth order Butterworth bandpass filter with upper and lower frequencies of 0.21 Hz and 0.28 Hz respectively was used to isolate the mode of interest. The natural frequency and damping ratio results using a curve fit are listed below. The damping ratio estimate 95% confidence interval for the curve fit is also reported in parentheses. The curve fit was applied over the first five, three, and then two cycles of the RD function.

• N^c= 5, f_y1= 0.254 Hz, ζ = 1.60% (1.57, 1.63)

• N^c= 3, f_y1= 0.254 Hz, ζ = 1.38% (1.34, 1.43)

0 2 4 6 8 10 12 14 16 18 20

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4

Time Lag, τ (s) AccelerationAmplitude,¨y1(mgn)

N_s= 7273

Fitted Model: x = Ae^−ζωt(cos(p

1− ζ²ωt) + B sin(p

1− ζ²ωx)) RD Signature Curve Fit

Figure 4.11: Ranked RD function for the fundamental x-axis translation vibration mode, generated from data obtained during the monitoring programme at Latitude tower.

• Nc= 2, f_y1= 0.254 Hz, ζ = 1.02% (0.98, 1.06)

4.5 Summary

This chapter described vibration models of structures and the system identification techniques used to determine dynamic characteristics from ambient vibration data.

Finite element models were initially discussed to introduce the formation of the equations of motion for discrete systems with multiple degrees of freedom, and to describe the solution methods for damped and undamped systems.

For the damped system case, proportional damping and general viscous damp-ing cases are discussed. The equations of motion for the general viscous dampdamp-ing case was used as the basis for the system identification techniques. Three system identification techniques were described: a frequency domain technique, and two time domain techniques. All three techniques estimate the dynamic characteristics via the response measurements from a structure excited by unmeasured random excitation — a technique also known as operational modal analysis.

The frequency domain technique described is known as frequency domain de-composition. It is an extension of the peak picking method, with the addition of

singular value decomposition of the spectral response matrix. Compared with the peak picking method, this additional step in FDD improves the estimates of dynamic characteristics, particularly damping ratios, and allows for improved identification of closely spaced modes of vibration. The first time domain system identification technique discussed is known as stochastic subspace identification. It is a paramet-ric technique that has similar advantages to FDD — namely the ability to separate closely spaced modes of vibration. The random decrement technique was finally in-troduced in order to determine the amplitude dependence of the natural frequency and damping ratio.

The FDD and SSI techniques were applied to wind-induced acceleration re-sponses recorded approximately two years after construction completion of Latitude tower. Natural frequencies and damping ratios for the first six modes of vibra-tion were estimated. The first six modes comprised three fundamental modes of vibration, including translation in the x-axis and y-axis and a torsional mode, and corresponding second modes of vibration for these fundamental modes. The natu-ral frequency estimates for the FDD and SSI techniques for all modes differed by less than 1%. The damping ratio estimates displayed more variability between the techniques. The damping ratio estimates for the fundamental modes differed by less than 28%, but the second modes displayed larger differences, with the SSI estimates being more than twice the FDD estimates for the second y-axis translation and second torsional modes.

Multiple system identification techniques were used to gain more confidence in the estimates of dynamic parameters. This is particularly relevant to the estimation of the damping ratios when using unmeasured ambient vibrations as the excitation force. The results from applying the system identification techniques to the vibration tests conducted during construction of Latitude tower are presented in the following chapter, in addition to comparisons with finite element models.

Chapter 5

Estimating Dynamic Characteristics

5.1 Introduction

The main focus of this chapter is the estimation of the natural frequencies and damping ratios of tall buildings. Results are presented for data obtained during both the construction cycle, and during the post construction stages with tenant occupation. A description of the tests and the equipment is included in Section 3.4.

The tests comprised forced and ambient vibration tests, and a comparison between the results from the two test types is presented.

The natural frequency results during construction are discussed with attention to the structural changes that occurred between consecutive vibration tests. The goal is to further the understanding of links between changes in the stiffness and mass of a structure, and the influence on the natural frequency. This will enable more accurate models of structures to be generated, which will improve the natural frequency estimates. In addition, a finite element analysis is also included as part of the natural frequency estimation process. Using the natural frequencies measured during the early stages of construction, a finite element model representing the struc-ture at the time of testing is updated in order to achieve improved accuracy between the model and the full-scale natural frequencies. The updated model attributes are then applied to a model representing the completed structure. The knowledge ob-tained from vibration testing, conducted during the early stages of construction, is expected to improve the natural frequency estimates from the finite element model of the completed structure. This is important for two reasons. Firstly, it provides an evaluation of the assumptions and techniques used in the design, and the out-comes of the evaluation can benefit future designs. Secondly, for buildings that require auxiliary damping, a more accurate natural frequency estimate will benefit the design of the damping devices.

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The damping ratio estimates during construction are discussed in parallel with the structural changes. For a complex structure such as a tall building, it is dif-ficult to attribute damping changes to particular changes in the structural and non-structural elements. The goal in this instance is to determine the change in the damping ratio between the initial stages of construction and after construction completion. A relationship between these damping ratio estimates would prove use-ful in removing much of the uncertainty in the damping ratio estimates for similar tall building structures.