Novel evaluation of mass polar moment of inertia of the drive plate in resonant column Open Access

Resonant column (RC) is a laboratory standard (ASTM, 2007) for the evaluation of dynamic properties of geomaterials such as shear wave velocity (V_s). The mass polar moment of inertia (MPMI) of the drive plate (I_o) is required for the solution of equation of motion in RC tests (e.g. Richart et al., 1970; Cascante et al., 2003). I_o is typically evaluated by standard procedure in which calibration probes of different resonant frequencies (f_o) and therefore different stiffness are tested.

In standard procedure, the evaluation of I_o is not consistent even if the probes are made out of the same material. The discrepancy in I_o is due to contribution from base of RC among other effects (Drnevich, 1978; Avramidis & Saxena, 1990; Khan et al., 2008; Clayton et al., 2009). The extent of contribution depends on the relative stiffness of the probe to that of the base. Amplitude of input excitation (current) and transfer function (TF) of peripheral instruments such as power amplifier can also affect the position of resonant frequency of the probes and therefore affect the I_o (Khan et al., 2013).

In this paper, a new method (NM) to evaluate I_o of drive plate is presented. In the proposed method drive plate is suspended with a thin polyamide wire and therefore the contribution from base of RC is absent. The response of suspended drive plate is governed by its inertia only because the damping coefficient (c) and stiffness (k) of suspension system is negligible. The validity of this assumption is verified by conducting free vibration of the drive plate and calculation of the torsional stiffness of the wire (K_w). The response of peripheral instruments does not affect the proposed method; however, amplitude of excitation has slight effect on the I_o. The results show consistent values of I_o at all values of frequencies within the tested bandwidth. The results also show that the apparent values of I_o from conventional procedure are higher compared to its inferred true value and the difference increases as the stiffness of the calibration probes increases. Consequently, the V_s of the soil specimen is overestimated for I_o values higher than its true value.

The specimen in an RC device under torsional excitation is modelled as a single-degree-of-freedom system (SDOF) if mass of the RC base is 100 times more than the mass of the specimen (Ashmawy & Drnevich, 1994) or MPMI of the base is 100 times larger than MPMI of the specimen (ASTM, 2007). Both conditions are difficult to fulfil in a typical RC device. Khan et al. (2013) showed that the base of the RC device shall not be assumed as fixed and that the apparent V_s shall be corrected depending on the ratio of torsional stiffness of the specimen (K_sp) to torsional stiffness of the base (K_b).

In standard procedure, the I_o of drive plate is evaluated by testing calibration probes (f_o = 20–100 Hz). The error in measured f_o of probes and therefore in I_o increases with increase in torsional stiffness of probes (e.g. Clayton et al., 2009; Sasanakul & Bay, 2010). Figure 1 illustrates the effect of f_o on I_o of drive plate from different studies. The curves in Fig. 1 are predictions from exponential models fitted to the original data to generate I_o values at similar values of f_o. The I_o and f_o values are then normalised to better visualise the trend. All studies indicate an increase of up to 45% in I_o with f_o ranging from 27 to 244 Hz. A similar percentage change (up to 20%) in V_s of a soil specimen is to be expected because V_s is proportional to I_o. Since the I_o values always increase with increase in stiffness of calibration probes; therefore, the true value of the I_o shall correspond to calibration probe of zero stiffness. The conventional procedure will therefore always result in an overestimation of V_s depending on the chosen value of apparent I_o. The effect of different values of I_o on the V_s of a sand specimen (Barco 71) is presented in Fig. 2. The sand specimen is tested at an isotropic confinement of 32 kPa and at shear strains smaller than 10⁻⁶. The I_o values in Fig. 2 vary from 0·008 to 0·0116 kg m², which represent the largest change (45%) in I_o reported in literature (Fig. 1). The V_s of the soil specimen consequently increases from 149  to about 180 m/s (increase of 21%). Figure 2 shows V_s values that are also corrected for the base effects in RC according to the procedure proposed by Khan et al. (2008). Considering the effect of f_o of probes on the I_o of drive plate and consequent error (over estimation) in V_s, an NM for the calculation of I_o is developed and presented in the following.

The displacement response (x) of a specimen under torsional excitation in RC device is presented in equation (1) (e.g. Cascante et al., 2005).

where F is the induced force in the coils due to current, J ( = I_o + I_s/3) is the weighted mass polar moment of inertia (MPMI) of the specimen (I_s) and driving plate (I_o), c is the viscous damping coefficient, r_m is the distance of magnets from the centre of the drive plate and r_a is the radial distance of the accelerometer from the centre of the drive plate. Derivatives of acceleration and velocity are represented by dots on the variables. Lorentz equation (F = BI_c) relates the magnetic-force factor (B) and current in coils (I_c) to yield force on the drive plate at a distance of r_m. In the absence of a specimen, k and c can be ignored, and J reduces to (I_o + ΔI_m). Equation (1) becomes equation (2) after rearrangement and substitutions for F and ẍ/r_a.

where α is the angular acceleration of the drive plate (ẍ/r_a). Equation (2) is an equation of line with slope equal to Br_m/(I_o + ΔI_m), dependent variable of α and independent variable of I_c. ΔI_m is MPMI of a calibration mass added to the drive plate. B and r_m are properties of coil–magnet system and do not change. The ratios of α and I_c are obtained by performing a sinusoidal chirp of frequencies ranging from about 30 to 200 Hz without and with addition of calibration mass, which results in slopes S₁ and S₂ as functions of frequency, respectively. Ideally, the slopes shall not change with frequency. The two slopes are then used for the calculation of I_o from equation (3) as a function of frequency.

Figure 3 presents the experimental set-up for the suspension of drive plate in RC along with the layout of instruments. The drive plate is attached to a three-dimensional (3D)-printed holder for levelling and a wire for lifting. The MPMI of the 3D-printed holder (6·5 × 10⁻⁷ kg m²) is very small which is later subtracted from the I_o. The other end of suspension wire is anchored to a point roughly 1 m above the drive plate. A signal generator, power amplifier, switch, filter amplifier, oscilloscope and spectrum analyser comprise of the main equipment set-up.

The wire is a made from monofilament polyamide with diameter (d_w) of about 0·5 mm. The shear modulus of the wire (G_w) is about 1150 MPa (manufacturer rating). The torsional stiffness (K_w) of the wire with length (l) is calculated from equation (4). The calculated K_w of the wire (7·1 × 10⁻⁶ N m) is very small compared to typical torsional stiffness of sand specimen at a confinement of 30 kPa in an RC test (around 1000 N m).

A sinusoidal chirp (30–200 Hz) is generated and amplified in power amplifier. The response of drive plate is measured with two accelerometers mounted at a distance of r_a from the centre. The current in the coils (I_c) and voltage signals from accelerometers are filtered (low pass 500 Hz) and amplified (20 dB) in the filter amplifier. The time signals are monitored in oscilloscope whereas the TF (ratio of acceleration (voltage) to I_c) is processed in real-time spectrum analyser. A uniform spectral window and ten averages are used to obtain the TF in frequency domain.

The TF values are multiplied by a factor to convert the acceleration voltages to angular acceleration for compatibility with Equation (2). The conversion factor is a function of sensitivity of accelerometer (mV/g), acceleration due to gravity (g), radial distance (r_a) and amplification (dB) if different from amplification of I_c signal. The converted TF values represent the slope (S₁) in equation (2). The test is repeated after addition of mass of known ΔI_m to the drive plate to obtain the slope S₂. Finally, I_o is computed from equation (3) as a function of frequency (30–200 Hz). TF approach is preferred due to its ability to eliminate noise due to spectral averaging and faster acquisition instead of repeating the test at different frequencies (alternate approach).

Figure 4 presents the free vibration decay of the drive plate, which is compared with the free vibration decay of a typical sand specimen from an earlier RC test. The drive plate is excited by a single frequency sinusoid. The switch (Fig. 3) stops the current to the coils and simultaneously triggers the oscilloscope for synchronous capture of free decay. The absence of plate vibration in Fig. 4(a) after termination of current confirms negligible stiffness of the wire and validates the assumption of neglecting k in equation (1).

The amount of phase difference between the input (current) and the response (acceleration) is proportional to damping ratio of the system (Khan et al., 2011). Input and response in a system with zero damping ratio will be either in phase or out of phase by π radians depending on the TFs of the instruments in the circuit. Figure 4(a) supports the elimination of c from equation (1) due to phase difference of +π radians between the current and acceleration signals.

Typical TFs between the acceleration response (voltage) and current are compared in Fig. 5. The variation of TF values within frequency bandwidth of interest is less than 0·005%. The current to the coils shall be kept constant during the sinusoidal chirp; however, fluctuations can occur due to response of the power amplifier. Fluctuation in current induced by the power amplifier at different frequencies is monitored by isolating the power amplifier and measuring its TF. The effect of variations in current on the ability of proposed method to evaluate I_o is insignificant. Figure 6 presents the TF of audio power amplifier used in this study. The proposed method performs very well in spite of using ordinary power amplifier which is not maintaining constant amplification of current at different frequencies.

Figure 7 presents the computed MPMI of the drive plate (I_o) as a function of frequency. The I_o values at two levels of average current in the coils (I_c) are compared. The average of the current is used to represent the two levels due to variations in current induced by power amplifier (Fig. 6). The upper level of current (0·2 A) in the coils represents high level setting, which is not typically used in calibration procedures. The effect is probably due to larger torsion in the drive plate and therefore higher contribution of K_w of the suspension wire. The change between smallest and highest values of I_o with frequency (27–108 Hz) at typical power level (0·05 A) is about 2·25%, which is much smaller than change in I_o values obtained from the probes (10–20% in this study). The change in I_o is about 1% if average value of I_o is considered. Figure 7 shows that I_o values tend to converge with decrease in resonant frequencies (stiffness) indicating that the apparent values of I_o from probes are higher than its true value.

Alternatively, variation of angular accelerations (α) and current in the coils (I_c) for a single frequency can be plotted to obtain the desired slopes. The plot yields a straight line with high r² values (>0·98) as shown in Fig. 8; however, this approach is not recommended. As noted in Fig. 7, the change in current level has an effect on I_o. A larger variation in current for the computation of slopes will induce some error in I_o.

RC can also be used as torsional-shear device at very low frequencies if the value of magnetic force factor (B) is known. Typically, this value is obtained through indirect measurements. The proposed methodology provides direct estimate of the magnetic force factor from equation (2). The magnetic force factor (B) of the tested RC device is 16·2 N/A.

A new methodology is presented in this paper to evaluate the MPMI of the drive plate of an RC device. Unlike the standard method, which relies on probes to evaluate MPMI and consequently suffer from many testing biases, the NM is based on suspension of drive plate with a thin wire of practically negligible stiffness and damping. Following important conclusions can be drawn from the study.

• The apparent values of I_o from conventional procedure are higher than its inferred true value and the difference increases as the stiffness of the calibration probe increases.

• The change in I_o values at frequencies of interest in the proposed NM is about 2%, which is much smaller than change of 10–45% from conventional procedure involving probes.

• The calculated V_s for a sand specimen increases (overestimation) by about 21% when I_o increases by 45% from its inferred true value.

• The TF approach in frequency domain is fast, eliminates noise in the signal by performing spectral averaging, and does not require individual measurements at different frequencies.

• The magnetic-force factor of the coils can be reliably computed from the proposed method, which is needed if a RC device is to be used as torsional shear device.

• The proposed method based on TF approach does not suffer from uncontrollable fluctuations of current from an ordinary audio power amplifier.

The authors acknowledge the support provided by Natural Sciences and Engineering Research Council of Canada (NSERC), the University of Waterloo, and in part, by the Open Access Program from the American University of Sharjah. The results and discussions are that of authors and do not reflect the opinion of institutions.

B

magnetic-force factor

c

viscous damping coefficient of the system

d

diameter of suspension wire

f_o

resonant frequency of probes

G_w

shear modulus of suspension wire

I_c

current in the coils

I_o

mass polar moment of inertia (MPMI) of drive plate

K_b

torsional stiffness of the base of resonant column

K_w

torsional stiffness of the suspension wire

k

stiffness of the system

l

length of suspension wire

r_a

distance of accelerometers from the centre of drive plate

r_m

distance of magnets from the centre of drive plate

V_s

shear wave velocity of soil specimen

α

angular acceleration of drive plate

ΔI_m

MPMI of the additional mass