Servo-hydraulic systems have complex dynamics and induce inevitable time-delay between the command and the achieved displacements. The actuator response is a combination of a delay and an amplitude error; however, they can be approximated as a simple but acceptable time delay system at lower frequencies that are mostly encountered in structural engineering real-time hybrid testing. Mercan and Ricles (2007) have found that the structural system is stable with an amplitude error in the
restoring force, whereas the time delay would potentially destabilize the system. The critical time delay is defined when the structural system becomes unstable. It is shown to depend on the test structural parameters. The structures with high damping or long period have a large critical time delay.
5.1.1 Pseudodelay Technique
Consider an elastic SDOF hybrid system that the displacement is measured from the test specimen with a time delay and an amplitude error. The inertia and damping
where ma, ca and ke are the analytical mass, damping coefficient, and elastic stiffness of the test specimen, respectively. The parameter λ is the ratio of the measured displacement to the command displacement. u(t), u&(t) , u&&(t) are the relative displacement, relative velocity and relative acceleration of the structure, respectively.
Peff(t) and τ are the effective load and the delay time, respectively. Eq. (5-1) can be represented in state space form as:
)
In Eq. (5-2), x(t), x&(t) are the state and state derivative vectors, respectively. x(t-τ) is the delayed state. As and Ad are the system matrix with no delay in the state and with delay in the state, respectively. Bs is the excitation input matrix, and w(t) is the external excitation. The characteristic equation of the delay system can be obtained by
( )
0 technique is adopted to transform the characteristic equation of the time delay system Ψ(s, e-sτ) into a polynomial q(Tr, s) by using an exact substitution for e-sτ proposed by Rekasius in 1980. The Rekasius substitution takes the form:+
The substitution replaces the infinite-order exponential function with the parameter Tr
converting Eq. (5-3) into a polynomial function:
0
Routh-Hurwitz stability test is adopted to efficiently solve the pairs of (Tcri, ωcri) that causes the instability of the structural system. The Routh-Hurwitz array of Eq. (5-6) is shown below:
column of the array is equal to the number of poles in the right-hand plan. It is noted that only the third element in the first column could be negative for some values of while the other elements remains positive. As a result, the critical value, Tcri, which causes the instability of the system, can be computed by solving the following equation:
0 )
2 ( )
(−λkeca Tr2− maλke−ca2 Tr +maca = (5-7)
where Tcri is the minimum root of the quadratic equation above. Once Tcri is obtained, substitute Tcri into Eq. (5-5) to find the corresponding ωcri by a pair of imaginary roots at s=±jωcri. Finally, substitute Tcri and ωcri into Eq. (5-5) to get the critical time delay, τcri.
For the case that Tr=0, the corresponding characteristic equation is Ψ(s,1) which is stable. For the case that Tr=∞, the existence of a pair of imaginary roots can be checked by examining the roots of Ψ(s,−1):
2+ a − e =0
as c s k
m λ (5-8)
Apparently, there imaginary roots exist if and only if ca =0 and ke <0 which is not occurred in real tests. As a result, for 0≤Tr≤∞, the critical time delay is τcri. Similar procedure is also discussed by Mercan and Ricles (2007).
5.1.2 Effect of Time Delay and Amplitude Error
The effects of time delay and amplitude error on the stability of the structural system are discussed by an example 2% damped structure. Table 5-1 shows the critical time delay associated with different λ values and structural periods. It indicates that when the structural period increases the critical time delay also increases. In addition, the critical time delay of the structure with a 5% damping ratio is larger than that of the structure with a 2% damping ratio, indicating that the when structural damping increases the critical time delay increases. The observation is consistent with the finding from an energy point of view (Horiuchi et al., 1999).
It has been demonstrated that an overshoot error adds a positive damping to the structural system (Mercan and Ricles, 2007). The critical time delay could have been increased when an overshoot error occurred as it would add a positive damping to the structural system. However, it is observed from Table 5-1 that a system with an undershoot amplitude error has a longer critical time delay than that with an overshoot amplitude error. This is contradictory to the finding that the critical time delay increases when the structural damping increases.
In fact, the structural period would be shortened when an overshoot error exists. The term λke in Eq. (5-1), can be viewed as an equivalent stiffness which is larger than the elastic stiffness ke. The combined effects of time delay and amplitude error can be investigated by Bode magnitude plots. Figure 5-1 shows the Bode magnitude plot of a 2% damped structure with a 3ms time delay and a 10% overshoot or undershoot amplitude error. The example structural properties are listed in Table 5-2 (Mercan and Ricles, 2007). It is observed that an undershoot amplitude error decreases the resonant frequency while an overshoot amplitude error reverses the result. In addition, a 3ms time delay increases the resonant peaks regardless the presence of the amplitude error.
Although an overshoot error adds a positive damping to the structural system as mentioned before, the resonant peak still increases when a time delay exists as shown in the enlarged plot of Fig. 5-1 Conclusively, although an overshoot amplitude error adds a positive damping to the structural system, its effect is like making the structural system stiffer. As a result, the critical time delay decreases when an overshoot amplitude error occurs in a time delay system.