# Optimal Design and Hybrid Control for the Electro-Hydraulic Dual-Shaking Table System
## 2 System Design and Optimization
### Hydraulic Power Supply Design  
For the hydraulic power supply of shaking tables, the flow and its change range is large [41; 44]. So, to reduce the cost, the hydraulic power flow is supplied by the hydraulic pump cooperating with the accumulators group, which is taken as the assistant hydraulic power supply. The average flow is supplied by the hydraulic pump and the flow exceeding the average flow is supplied by the accumulators group.  
The shaking tables system is mainly used to perform earthquake simulation tests, so the flow of hydraulic power supply is usually designed according to El-Centro acceleration waveform, which is a typical earthquake waveform recording a strong earthquake. In the reference [40], the suggested flow of the hydraulic power supply for the shaking table system is \(0.1Q_{\max}\), which is the statistical calculation result of the three typical earthquake waveforms: El-Centro, San-Formando and Parfield. For the shaking table system researched in this paper, the flow \(0.1Q_{\max}\) is only about \(260\) L/min, which is relatively small and only considers the earthquake waveform experiment. The needed flow for the El-Centro waveform is a half of the flow for the uniform random waveform. So, for more common use in the uniform random waveform experiment, this paper designed the flow according to the El-Centro earthquake waveform, whose maximum velocity is twice the required one in Table 1. Note that the design of the flow only uses the waveform of the El-Centro, the quantity of the maximum velocity is based on the requirement of the system. The El-Centro acceleration and velocity waveforms are shown in Figure 7a,b.  
Then, the average flow of the horizontal hydraulic cylinder can be obtained  
\[\overline{Q}=\frac{\int_{0}^{T}Q_{h}(t)dt}{T}=\frac{\int_{0}^{T}|A_{x,y}\cdot \int_{0}^{t}a(t)\cdot dt|dt}{T} \tag{9}\]  
where, \(T\) is the total time of El-Centro waveform, \(a(t)\) is El-Centro acceleration waveform.  
According to Equation (9), we can obtain the total average flow as  
\[\overline{Q_{t}}=2\overline{Q}_{x}+2\overline{Q}_{y}+4\overline{Q}_{z}=458\) L/min (10)  
Considering the oil supply of pilot valve, we choose the flow supplied by the hydraulic pump as \(Q_{B}=500\) L/min for each shaking table.  
The working volume of the accumulators group at time \(t\) is  
\[V_{w}(t)=\int_{0}^{t}\left(Q_{h}(t)-Q_{B}\right)\cdot dt=\int_{0}^{t}\left(|A_ {x,y}\cdot\int_{0}^{t}a(t)\cdot dt|-Q_{B}\right)\cdot dt \tag{11}\]  
Figure 7: The El-Centro waveform curves. (**a**) The acceleration curve; (**b**) The velocity curve.  
The maximum change of \(V_{w}(t)\) is the needed working volume of the accumulators group. From Equation (11), the maximum total change \(V_{wc}\) for horizontal and vertical \(V_{w}(t)\) is  
\[V_{wc}=4V_{wch}+4V_{wc}=63.6\;\mathrm{L} \tag{12}\]  
In this case, the accumulators group is used as the assistant hydraulic power supply, so the velocity of the accumulator to charge and drain oil is very fast, and the volume change can be computed according to the adiabatic formula. Combined with Equation (12), the volume of the accumulators group is [41]  
\[V=\frac{V_{wc}\left(p_{h1}/p_{h0}\right)^{\frac{1}{n}}}{1-\left(p_{h1}/p_{h2} \right)^{\frac{1}{n}}}=506.2\;\mathrm{L} \tag{13}\]  
where \(P_{h1}\) is the minimum working pressure, \(P_{h1}=0.8P_{h2}\); \(P_{h2}\) is the maximum working pressure, \(P_{h2}=P_{s}\); \(P_{h0}\) is the pre-charged pressure, \(P_{h0}=0.8P_{h1}\) and \(n=1.4\).  
In conclusion, the flow of the hydraulic power supply of the shaking tables system supplied by the hydraulic pump is 1000 L/min and the volume of the accumulators group is 1000 L.