## 3 Stiffness Analysis for ALS-EHA
### Frequency Characteristics of Stiffness  
For a sinusoidal disturbance load with frequency \(w\), the magnitude of the transfer function is. \[\left|\frac{x_{a}(iw)}{F_{l}(iw)}\right|=\frac{\left[\left((Mw)^{2}-MK_{h}+C^{ 2}\right)^{2}w^{2}+\left(CK_{h}\right)^{2}\right]^{\frac{1}{2}}}{wK_{h}\left( (Mw)^{2}+C^{2}\right)} \tag{25}\]  
Equation (25) represents that at low frequencies band, as the frequency \(w\) reaches zero (DC input), the displacement \(x_{a}\) tends to infinity. This means that, in the absence of control, the load will be able to rotate the motor freely until the actuator reaches either end of the hydraulic cylinder. At high frequencies, the magnitude \(\left|x_{a}(jw)/F_{l}(jw)\right|\) decreases to \(1/K_{h}\) as frequency \(w\) increases. This is because the equivalent mass of the motor-pump inertia does not respond to high frequencies. Therefore, the ALS-EHA stiffness at this time is the hydraulic stiffness of the oil in the cylinder. the ALS-EHA control system can determine its stiffness within its bandwidth range. In fact, it should be paid the most important attention to the stiffness of the frequencies between the EHA bandwidth and the high frequencies, where \(K_{d}\) denotes the stiffness, which is identical to the one in valve-controlled actuators. Equation (24) can be reorganized as:  
\[\frac{x_{a}(s)}{F_{l}(s)}=\frac{s^{2}+2\alpha_{n}\xi s+\alpha_{n}^{2}}{(s^{2}+ 2\alpha_{n}\xi s)K_{h}} \tag{26}\]  
where \(\alpha_{n}=\sqrt{K_{h}/M}\) is undamped natural frequency, \(\xi=C/2\sqrt{K_{h}M}\) is damping ratio. Equation (26) can be simplified as follows:  
\[\frac{x_{a}(s)}{F_{l}(s)}=\left(\frac{1}{K_{h}}\right)\!\!\left|1+\frac{\alpha _{n}^{2}}{(s^{2}+2\alpha_{n}\xi s)}\right| \tag{27}\]  
When \(F_{l}\) a sinusoidal disturbance of unit amplitude, the steady state response of output displacement of the actuator is:  
\[x(jw)=\left(\frac{1}{K_{h}}\right)\!\!\left|1+\frac{1}{\tau(-\tau+\jmath 2 \xi)}\right| \tag{28}\]  
\[\left|x(jw)\right|=\left|\left(\frac{1}{K_{h}}\right)\!\!\left|1+\frac{1}{\tau (-\tau+\jmath 2\xi)}\right|\right| \tag{29}\]  
\[=\left.\left(\frac{1}{K_{h}}\right)\!\frac{\left[\left(\tau^{3}+4\tau\xi^{2} -\tau\right)^{2}+4\xi^{2}\right]^{\frac{1}{2}}}{\tau(\tau^{2}+4\xi^{2})}\right.\]  
where \(\tau=\frac{w}{w_{n}}\) is the normalized frequency. The inverse of \(\left|x_{a}(jw)\right|\) is the stiffness \(K_{d}\) of ALS-EHA to a unit amplitude disturbance load without the control. Figure 11 shows the relationship graph of the normalized stiffness \(\kappa=K_{d}/K_{h}\) versus the normalized frequency \(\tau\) for different hydraulic reduction ratios. it shows that, as the frequency decreases at low frequencies, the stiffness gradually approaches zero, while at high frequencies, the stiffness for hydraulic reduction ratios is close to \(K_{h}\), for which the most obvious difference is at the natural frequency point. For high hydraulic reduction ratios, the stiffness tends to be high at the natural frequency. In this mode, the disturbance causes the motor to rotate a large angular displacement while the load remains stationary. As the hydraulic reduction ratio falls, the stiffness at natural frequency decreases and tends to \(K_{h}\). What's more important is that the stiffness of ALS-EHA with a low hydraulic reduction ratio increases much faster at low frequencies than that of the high hydraulic reduction ratio one. EHA dynamic stiffness characteristics is commonly adopted to guide its design. For the purpose of ensuring the stable stiffness of ALS-EHA, one method is to design the ALS-EHA bandwidth higher than the natural frequency of the hydraulic spring and equivalent inertia of the rotating components (mass -spring). This method may be problematic that the high bandwidth of ALS-EHA involves some structural modes leading to it actively responding to these structural vibrations, maybe inducing system instability.