EFT_Stage2_Details

Test structure

The structural model consisted of a concrete mass atop four caster wheels with two springs on each side of the structure in the direction of motion.  The concrete mass weighed 15.5 kip.  The springs were designed to lose contact with the mass at displacements exceeding 1 in., resulting in a reduced stiffness.  Thus, the structure was a linear elastic structure with an initial stiffness of 4 kip/in. when the displacement response was within the 1-inch precompression, while it acted as a nonlinear elastic structure when the displacement response exceeded the precompression (the stiffness reduced to 2 kip/in..  An automobile suspension strut was used as a damper in some tests while a fluid viscous damper was used in others.  with the viscous fluid damper, a structural damping of 3.0% of critical damping was obtained through fitting the free vibration test results.

Connection with the springs and actuator

   Springs were type UM-1000 from Belts Spring Company, San Leandro, CA with a nominal stiffness of 1 kip/in. and 6-inch travel length.  The 1-inch precompression was enacted as follows: starting from the null position of the actuator piston, the actuator was commanded to have positive 1 inch offset (the cart was pulled back by 1 inch).  The connections of the springs on the opposite side were adjusted such that the springs touched the mass.  Then the actuator was commanded to have negative 1 inch offset, and the connections of the springs at the actuator side were adjust such that that the springs touched the mass.  Finally, when the actuator went back to its null position, all springs were compressed by 1 inch.  A maximum cart (mass) displacement of 4 in. was set in the servovalve controller to avoid damage of the springs due to extensive deformation.

Working on concrete pouring

The natural velocity feedback is shown by the loop (a)-(b)-(c) from the structural velocity to the summing point (c), which represents the law of conservation of mass: the hydraulic flow into the actuator needs to counteract the fluid compressibility, system leakage, and chamber volume change. The effect of the natural velocity feedback loop (a)-(b)-(c) is compensated by a positive feedback loop (a)-(d)-(e)-(f). In order to cancel the effect of the natural velocity feedback at point (c), the compensation loop needs to incorporate the inverse of the dynamics between (f) and (c) (forward dynamics), which represents the behavior of the servovalve and its controller. Therefore, the servovalve and its controller need to be characterized in detail.

The mathematical models of the servo-system have been derived based on the formulations by Merritt (1967).  The dynamics of the three-stage servovalve (H_fd) contain three major components:

(1) the proportional-integral-derivative (PID) control with zero I gain,

where G_p and G_d are the proportional and derivative gain of controller, respectively;

(2) the second-order servovalve dynamics,

where t is the equivalent time constant of the pilot-stage valve, K_vp is the pilot-stage valve flow gain, A_v is the main-stage spool area, K₃ is the sensitivity factor of the internal LVDT, and x_vmax is the maximum spool stroke;  and

(3) the nonlinear servovalve flow characteristic stated by

where x_v is the spool opening of the servovalve (-1 to 1),  K_v is the no-load flow gain of the servovalve, which is a function of spool opening, P_L is the load pressure (P_LA is approximately the force applied to the structure, and A is the actuator piston area), and P_s is the supply pressure.

Nonlinear servovalve flow gain (K_v)

A flow curve (i.e., the flow controlled by servovalve Q_L vs. the main-stage spool opening x_v) was constructed based on the law of conservation of mass

where K_a is the compressibility coefficient of the hydraulic fluid inside both actuator chambers, and C_l is the total leakage coefficient of the servovalve/actuator combination. The actuator was put into displacement control with a unity controller P-gain and zero D-gain.  Tests were conducted under no load condition (the structure was disconnected) such that the pressure difference across the actuator piston (load pressure P_L) was negligible.  The leakage flow was neglected because the leakage was typically small (less than 0.5% of valve capacity).  The spool opening was obtained directly by measuring the main-stage spool position while the corresponding flow was calculated as the piston velocity multiplied by the piston area.  The piston velocity was calculated using the central difference method from the measured piston displacement.

Test result with the 90% full stroke command is shown below. A piecewise linear curve that connected 21 control points at intervals of 10% of the spool opening was constructed to represent the flow property of the servovalve.

Servovalve response delay

The servovalve response delay was determined using the second-order servovalve model, which requires many valve parameters such as valve spool area and the maximum spool stroke.  The valve parameters are typically not all available; hence, a measured frequency response was used to estimate the parameters for an equivalent second-order model shown below, from which the the response delay was estimated.

Controller gain setting

Relatively large P gains should be used because they usually improve the overall performance of a stable system.  On the other hand, larger controller P gains may cause instability, and result in a high-frequency vibration of the actuator.  The maximum controller P gain can be obtained through trail and error; however, a stability analysis of the test system could be used to provide a guideline.  A unity P gain was used in this study.

To inverse the nonlinear servovalve flow characteristic, the chamber volume variation to be compensated was first multiplied by

to consider the effect of large forces applied to the structure (load pressure influence).  This process required two more inputs, the spool opening (x_v) and the load pressure (P_L).  The spool opening was obtained directly from the servovalve controller while the load pressure was approximated by the applied force divided by the piston area.  Secondly, the nonlinear no-load flow gain (K_v) was represented by a piecewise linear curve (servovalve flow vs. spool opening); hence, the inverse relation was simple once the flow curve was identified:  A look-up table based on the piece-wise linear flow curve was used to find the required spool opening to compensate the flow to the actuator.

The direct inverse of the servovalve dynamics results in a transfer function with a second-order term in the numerator, corresponding to an inherently unstable system because it can greatly amplify signals with high frequencies, such as electrical noise.  For the frequency range of interest (i.e., 0-10 Hz in this study), the servovalve dynamics was represented with reasonable accuracy by a first-order delay (K_s/(T_ds+1), where K_s is the valve gain) with a time constant (T_d); hence, a first-order phase-lead network was used to invert the valve dynamics.

where the constant a was taken as 0.1 because it could provide both good phase-lead performance (the performance would be reduced if a were too large) and acceptable noise amplification (noise would be greatly amplified if a were too small).

PID control with a zero I gain introduces some phase lead into the DC error signal if the derivative gain (controller D gain) is not zero.  Because the D gain was usually set very small (e.g., 0.2 ms) in this study (systems with large D gain would amplify noise signals), the controller dynamics were simplified as a pure gain, and the related phase lead was lumped into the dynamics of the servovalve: the lead-time (G_d/G_p) was considered by reducing the servovalve response delay (T_d).  Hence, the inverse dynamics of the servovalve controller was simply 1/G_p.

• Velocity was measured using a Velocity Transducer.
• Operational amplifiers (LF353N) were used to make the unit buffer, the phase-lead network, and the summing circuit