# Determination of critical timing of impulses

Consider the Input Sequence 2 (see Figure 4.2) in this section. The time interval between two consecutive impulses can be obtained by solving the equations of motion and substituting the continuation conditions at the transition points. The time interval *t _{0}^{5}* between the first and second impulses and the time interval

*t*between two consecutive impulses after the second impulse can be expressed by

_{0}^{c}where *t _{OA},* f

_{AB},

*t*f

_{6C}, t_{CD},_{DE}, and

*t*are the time intervals between two consecutive transition points shown in Figure 4.3(b). If

_{Ef}*V/V*< 2 is satisfied, r

_{y}_{0}

^{5}/T, = 1/2 holds. On the other hand,

*t*is obtained from Eq. (4.7b) in all input velocity level. The time intervals

_{0}^{c}*t*t

_{OA}, t_{AR},_{BC},

*t*t

_{cu},_{DE}, and f

_{EF}are expressed by

In Eqs. (4.8a—f), V denotes the normalized velocity input *VIV.*

Figures 4.5(a), (b) show two normalized time intervals *t _{0}^{s}/T_{x}* and t

_{0}

^{f}/T, with respect to the input velocity level

*VIV*These time intervals coincide with the time intervals between the points corresponding to the zero restoring force (see Figure 4.3). The timing is delayed due to the existence of plastic deformation as the input level increases. Only the critical response giving the maximum value of

*ujd*is sought by the method explained in this chapter, and the critical resonant frequency is obtained automatically without repetition for the increasing input level of the multiple impulse. This idea is almost the same as in the case of the double and triple impulses. One of the original points in this chapter is the tracking of only the critical elastic- plastic response.

_{y}*Figure 4.5* Normalized timing t_{0}^{!}/T, and t_{0}^{c}/T, with respect to input level: (a) *t _{0}4T*

_{h }(b) t

_{0}

^{c}/Ti (Kojima and Takewaki 2015c).

# Correspondence of responses between input sequence 1 (original one) and input sequence 2 (modified one)

Figures 4.6 and 4.7 present the time-histories of relative displacement (relative to the base motion), relative velocity and restoring-force, and the force- deformation relation under Input Sequence 1 with the resonant impulse interval *t _{0}/to* = 1.0 for the input velocity levels

*V/V =*2 and V/V = 3, respectively. In Figures 4.6 and 4.7, the structural parameters

*w*2^[rad/ sec] (T, = 1.0[sec]) and

_{x}=*d*= 0.16[m] are used. Since the steady state is very sensitive to the time increment in the time-history response analysis using an EPP model, the time increment was chosen as 1.0xl0-

_{y}^{6}[sec]. This phenomenon is well known as plastic flow. In fact, an EPP model was not treated in most works (Caughey 1960a, b, Iwan 1961,1965a, b) for its difficulty in the treatment of unstable and input-sensitive states. It should be remarked that the impulse timing is the critical one obtained by using the Input Sequence 2. The circles in Figures 4.6 and 4.7 present the acting points of impulses. It

*Figure 4.6* Response to Input Sequence I for *V/V _{y} =* 2 (impulse timing is critical one obtained by using Input Sequence 2): (a) displacement, (b) velocity, (c) restoring force, (d) force-deformation relation (Kojima and Takewaki 20 I 5c).

*Figure 4.7* Response to Input Sequence I for *VIV _{y} =* 3 (impulse timing is critical one obtained by using Input Sequence 2): (a) displacement, (b) velocity, (c) restoring force, (d) force-deformation relation (Kojima and Takewaki 20 I 5c).

can be seen that, although some irregularities appear at first, the response converges gradually to a state with the acting timing of impulse at the zero restoring-force point irrespective of the input velocity level.

Figure 4.8 illustrates the force-deformation relation under the multiple impulse of Input Sequence 1 with three time intervals *t _{0}/t^{c}0* = 0.8,1.0,1.2 for two input levels V/V, = 2, 3. While, in Figures 4.6 and 4.7, only the case of

*t*1.0 was treated, three cases

_{0}/to =*t*= 0.8, 1.0,1.2 of impulse time intervals are investigated in Figure 4.8. It can be assured that the response converges to a steady state irrespective of the impulse timing and the time interval

_{0}/t^{c}0*t*=1.0 certainly provides the maximum plastic deformation amplitude

_{0}/t‘_{0}*u*Furthermore, the time interval

_{p}.*t*=1.0 converges to the applied times of multiple impulses corresponding to the zero restoring force (both directions at counter locations). This indicates the validity of introducing the Input Sequence 2 for finding the critical timing of multiple impulse even for Input Sequence 1.

_{0}/t^{c}0*Figure 4.8* Response to Input Sequence I for three impulse timings (f_{0}/fo = 0.8, 1.0, 1.2) with two input levels *VIV _{y}=2,* 3 (Number beside circle: acting sequence of impulse) (Kojima and Takewaki 201 5c).

On the other hand, Figures 4.9 and 4.10 present the time-histories of relative displacement to the base motion, relative velocity and restoring-force together with the force-deformation relation under Input Sequence 2 for the input velocity levels *V/V _{y}* = 2 and

*V/V*3, respectively. In Figures 4.9 and 4.10, the structural parameters

_{y}=*w*= 2^[rad/sec]

_{x}*[T*= 1.0[sec]) and

_{x}*d*0.16[m] were used. The response exhibits a steady state from the initial stage and corresponds to a state with the timing of impulse at the zero restoring-force point irrespective of the input velocity level. It is seen that the critical timing

_{y}=*to*of multiple impulse computed by Eq. (4.7b) can be obtained without repetition and can be used as the critical timing even for the Input Sequence 1.