Pseudo‐active actuators: A concept analysis

The superior performance of active vibration control systems largely depends on the four‐quadrant controllable execution capability in the available force‐velocity diagram of active actuators. Although semi‐active vibration control systems have the advantages of low energy consumption, simple structure, and high reliability, the system performance is not comparable to active control systems, due to the partial capability in only the first and third quadrants. On the basis of the comprehensive advantages of active and semi‐active actuators, to reform the design philosophy of semi‐active actuators to realize pseudo‐active actuators that have both mechanical properties of active actuators and energy consumption advantages of semi‐active actuators, that is, new semi‐active actuators with four‐quadrant controllable execution capability, will very likely cause a revolution in the related fields of mechanical design and system control. The basic design principle of pseudo‐active actuators that use semi‐active controllable actuators to achieve active actuator performance in the way of conceptual analysis is proposed. The proposed pseudo‐active actuators should consist of two half‐four‐quadrant actuators, that is, one is for the first and third quadrants and the other one for the second and fourth quadrants. This study employs two semi‐active controllable damping actuators and one mechanical compensation mechanism. One of the actuators provides the damping force in the first and third quadrants, and the other one combining with the mechanical compensation mechanism is for the second and fourth quadrants. A global mathematical model of the proposed actuator is established to describe the four different operational modes of the proposed actuator. It is proved that the two operational modes of the proposed actuator can realize active vibration control, and a case study of realizing active control is presented. The other two operational modes are compared with the conventional two‐degree‐of‐freedom model. More specifically, the application cases of the pseudo‐active operational mode of the proposed actuator in the quarter‐car/body‐powertrain suspension system are given, a pseudo‐active suspension named dual‐hook automobile suspension is presented. Furthermore, an equivalent expression of the electrical network is given for the mechanical network under different operational modes of the proposed actuator.


| INTRODUCTION
Efficient suppression or utilization of mechanical vibration can avoid unwanted results, such as fatigue of the structural parts of the system and reduced comfort/safety performance. [1][2][3][4][5] Although the active vibration control system has superior performance, it encounters coldness in the process of large-scale promotion due to its "congenital defects," that is, huge energy consumption, bulky system, and high cost. In contrast, the semi-active vibration control system receives very high expectations due to its low energy consumption, simple system, and high reliability. 6 However, typical semi-active shock absorbers with adjustable damping/stiffness/inertial capacity, for example, electromagnetic/ magnetorheological shock absorbers, regenerative shock absorbers, controllable valve-type shock absorbers, air springs, and inverters cannot provide negative forces as active actuators do. On the one hand, the reason is that active actuators can achieve mechanical control within the four quadrants of the available force-velocity diagram, while semi-active actuators can only achieve a limited region in the first and third quadrants of the available force-velocity diagram, 7,8 as shown in Figure 1. More essentially, from the energy point of view, the active control method and the passive control method have different energy flow directions. The output mechanical energy of active actuators in the control loops could be from other forms of energy, such as electrical energy 9,10 and hydraulic energy. 11 In contrast, in the semi-active control systems, the semi-active actuators are the same as the passive actuators, which can only accept mechanical energy. 12 Other forms of energy input to semi-active actuators for control 13,14 are not converted into external mechanical energy. Therefore, in the full frequency range, taking the automobile ride comfort indices as an example, there is a nonnegligible theoretical gap between the semi-active suspensions and the active suspensions. The root-mean-square value of the former is lower than about 20% of the latter. 15,16 Figure 1 also presents the inability of the semi-active actuators for control systems. The existing active/semiactive vibration control technology has failed to change the state of "stalling" in the large-scale promotion of intelligent suspensions, and the reason would be the "congenital defects" of the active/semiactive suspensions.
The boundary between the concepts of active actuators and semi-active actuators is clear, but the boundary between the corresponding control strategies is ambiguous. Semi-active control systems based on active control strategies and semi-active actuators show similar characteristics of control effect as active actuators. It can be inferred that the realization of active control does not necessarily require active actuators. It is an important scientific concept and idea to study how to use semi-active actuators, e.g., controllable stiffness/damping/inertance actuators, with new topological relationships to achieve pseudo-active actuators with active control effects, that is, semi-active actuators with four-quadrant controllable execution capabilities. Again, taking automotive suspensions as an example, it is conceivable to realize a new type of intelligent suspension by using pseudo-active actuators, which not only has the performance of active suspension but also avoids its "defects" of high energy consumption and high cost. It also has the advantages of simplicity, reliablity, and low energy consumption of the semi-active suspensions and can avoid the lack of performance in the high-or low-frequency range. It may be an important way to promote a new round of research and promotion for automotive intelligent suspensions as well as other vibration/shock control.
Negative stiffness, with the reactive force to assist deformation progress instead of resisting, [17][18][19] is one of the most wonderful elements to implement the output force transition. A schematic of the force transition of a controllable damping actuator via a linear

| CONCEPT AND PRINCIPLE
The concept "pseudo-active actuator" is defined by the actuators that use semi-active controllable actuators to achieve active actuator performance. The pseudo-active actuators are then expected to possess both advantages of semi-active actuators-low energy consumption, simple structure, and high reliability, and of active actuators-four-quadrant execution capability as well as superior system control performance. Inspired by the performance analysis of a negative stiffness damper, as seen in Figure 2, we propose a basic design principle of the pseudo-active actuators in Figure 3. Observing Figure 3, the proposed pseudo-active actuator should consist of two half-four-quadrants, which may be implemented via two mechanisms.
One is for the first and third quadrants and the other one is for the second and fourth. As seen from Figure 3, this study employs two semi-active controllable damping actuators as an example and one mechanical compensation mechanism. 28 One of the actuators provides the damping force in the first and third quadrants, and the other combining with the mechanical compensation mechanism is for the second and fourth quadrants. As reasonable, it is inferred that pseudo-active actuators are also possibly realized via any combination of positive controllable stiffness/damping/inertance and mechanical compensation mechanism. Here it is noted that the combination of the mechanical compensation mechanism and another (controllable) stiffness/damping/inertance is used to provide (controllable) negative stiffness/damping/inertance in the second and fourth quadrants. For sure, if the second-and fourth-quadrant available outputs can be realized by specific actuators, such as permanent magnet, electromagnetic device, or others, then the mechanical compensation mechanism is not needed anymore.
F I G U R E 2 Performance of the available force of a specific pseudo-active actuator: negative stiffness + controllable damping According to the design principle presented in Figure 3, proposed pseudo-active actuator, respectively. This study takes the controllable damping for the pseudo-active actuator as an example, and any combined realization of controllable damping/controllable stiffness/controllable inertance could be implemented under the proposed principal frame. According to Figure 4A, the pseudoactive actuator is composed of two semi-active actuators and a rackpinion motion pair. Four different structural forms between the two rigid bodies and the two semi-active actuators can be completed with application requirements. In Figure 4A, y o , y g , and y u are, respectively, the displacements of the output end, the ground end, and the input end, and these three components of this pseudo-active actuator are where c ̅ l and c l are the damping coefficients with the same absolute value while opposite mathematical sign; k̅ l and k l are the stiffness coefficients with the same absolute value while opposite mathematical sign.
According to the d'Alembert principle, the force analysis of the grounded gear z g in Figure 4 can be expressed as m y r rF For the left rack z l , its force F l is (iv) Mode IV: when S 0 is off and S 1 on, the actuator has a single DOF.
In this mode, there is no relative movement between the two masses.
Regardless of the internal parameters of the mechanism system, it shows a damped single DOF to the outside. This study will not make a specific analysis of its characteristics.

| Proof of feasibility of active control
As shown in Figure 4B, when the switches S 0 and S 1 are both on, the According to Equation (6a), when S 0 and S 1 are both on, the force F mo on m o is determined by four parameters, that is, the damping coefficient difference c c − r l , the stiffness difference k k − r l , the input velocity yu, and the input displacement y u . The force F mo on m o is the output as the desired force F des , and the desired force output can be achieved by changing the four parameters. Among the four parameters, only the damping coefficient difference c c − r l can be tuned arbitrarily, and the stiffness difference k k − r l is set as a known quantity. Mathematically, for any desired force F des at any time, as long as the inputs y u and yu are given, the required damping coefficient difference c c − r l can be obtained. It is given by Smith defines passive elements as 12 where t is the continuous time; f(t) is the force on the two ends of the element, and the positive direction is the compression direction of the element; v(t) is the velocity at the two ends of the element, and the positive direction is the direction in which the two ends of the element approach each other.
According to Equation (9), for passive elements, mechanical energy can only flow into the element from the outside, but the element cannot output mechanical energy to the outside. For semiactive elements, it is essentially passive elements with adjustable parameters/performances. So Equation (9) will also describe the characteristics of the semi-active elements. Further for passive damping elements, Equation (9) is strictly greater than zero.
The value range of Equation (9) is an entire real number range for active elements. Figures 1A,B are the working ranges of semi-active (passive) actuators and active actuators, respectively. That is, the working range of active actuators is the entire plane, while the working range of semi-active (passive) actuators is only partial first and third quadrants, that is, Q1 and Q3.
If S 0 is on and S 1 off, and the mass m o is set to zero, the input end On the basis of the above analysis, it can be concluded that the mechanical performance of the active actuator as shown in Figure 1B can be realized in theory by using the semi-active actuators with a mechanical compensation mechanism. Combining Equation (6a) and the assumption above yields  Figure 5A shows the simulation results of the active-controlled output and Figure 5B shows those of the output frequency control effect. Excitation input was assumed to be y t = sin(1 ) u in the simulation.

| Case study of active control
According to Figure 5A, the pseudo-active actuator can use semi-active actuators to achieve an active-controlled output force in this mode, which is different from the excitation frequency. It can be seen from Figure 5B that the controlled rigid body m o is able to move at a frequency different from the excitation frequency. It indicates that the pseudo-active actuator has the potential to achieve more complex active control effects. two equivalent components must be connected to the ground. After disconnecting the common port from the ground, that is, both S 0 and S 1 are off, the pseudo-active actuator is in a 2DOF operational mode, i.e., Mode III. As shown in Figure 4B, it is found that the pseudo-active actuator has two elements with negative stiffness k̅ l and negative damping c ̅ l in the 2DOF operational mode. They are introduced by the mechanical compensation mechanism after the new topological construction. This new 2DOF model is different from the widely accepted conventional 2DOF model in Figure 6, in the topological relationship between the components and the characteristics of the used components. As seen from Figure 6 (2) and (5) where the subscript "n" represents "new". Incorporating Equations (14) and (15) The dimensional and nondimensional parameters of Equations (14), (15), (17), and (18) where the subscript "c" represents "conventional".
Similarly, the nondimensional method is used to simplify the calculation: Incorporating Equations (21a)-(21g) into Equations (19) and (20) The dimensional and nondimensional parameters of Equations (19), (20), (22), and (23) are listed in Table 2. For the mass ratios μ n and μ c , the influence on the response of the new 2DOF model is shown in Figure 9, and that on the conventional 2DOF model in Figure 10. Furthermore, since the two controllable damping actuators in the two 2DOF models are connected to m 1 (or m g ), both the damping ratios ξ n1 (or ξ c1 ) and ξ n2 (or ξ c2 ) influence the response of m o (or m 2 ).
Thus analysis of the two 2DOF models is conducted with the combined considerations of the damping ratios ξ n1 (or ξ c1 ) and ξ n2 (or ξ c2 ). Figure 11 shows the influence of the damping ratios ξ n1 and ξ n2 on the frequency response of the new 2DOF model. The influence of the damping ratios ξ n1 and ξ n2 on the gain at the two natural frequencies of the new 2DOF model is shown in Figure 12. Again, for comparison and analysis, the influences of the damping ratios ξ c1 and ξ c2 on the frequency response and the gains at the natural frequencies of the conventional 2DOF model are presented in Figures 13 and 14, respectively. Figure 11 shows that the frequency responses of m o and m g are similar to those in a single-DOF system. It can be seen from Figure 11A that ξ n1 has no obvious effect on the gain at any frequency of m o . In Figure 11B, for the new 2DOF model, ξ n1 only affects the gain of m g at its natural frequency. In Figure 11C, ξ n2 has a significant effect on the gain near the natural frequency of m o , but little influence on the gain at the first-order natural frequency of m g . As ξ n2 increases, the gain at the second-order natural frequency of m g decreases. But when ξ > 0.2 n2 , the response of m g changes little, as shown in Figure 11D. In Figures 14A,B, the angles between the gradient lines and the coordinate axis are large, and the effects of ξ c1 and ξ c2 are coupled to the responses of m 1 and m 2 . It is worth noting that the response of m 2 in Figure 14A is negative, while the Z-axis coordinate scale is also different from those in Figures 12 and 14B.

| Influence of nondimensional parameters on the transient performance of the system
As seen in the parameters listed in Tables 1 and 2 . Each parameter takes a value from Abbreviation: 2DOF, two degrees-of-freedom. 1e − 3 to 1e3, whose scope is sufficient to cover most engineering applications. The parameter-system pole diagram is shown in Figure 15.
It is known that the pole closest to the imaginary axis and near-zero is the dominant pole of the system. It has a large time constant and a slower settling time, which plays a leading role in the time domain response of the system. Therefore, the stability of the system should be compared with the real part of the pole closest to the imaginary axis. In Figure 15A, the new 2DOF model has only imaginary complex roots for the root locus of the changes of α n and α c , while the conventional 2DOF model has real roots. In contrast, in Figure 15B, the new 2DOF model has real roots for the root locus of the changes of μ n and μ c , while the conventional one has only imaginary complex roots except for the zero poles. As observed in Figure 15C, when ξ > 0 n2 , the stability of the new 2DOF model is greater than that of the conventional one near the default parameters. The new 2DOF model is more stable than the conventional one. When ξ < 0 n2 , as shown in Figure 15D, the roots of the new 2DOF model are mostly distributed on the right half plane. The system is unstable. For the conventional 2DOF model, no matter how the nondimensional parameters change, the roots of the system are distributed on the negative half plane. It indicates that the system is stable. Hence a stable parameter set of the new 2DOF model system is given to ensure system stability in practical applications.

| Conditions for system stability
To ensure the stability of the new 2DOF model in practical applications, it is of significance to find the corresponding parameter set.
The Routh-Hurwitz criterion is a necessary and sufficient condition for system stability. Conditions for the Routh-Hurwitz criterion of the fourth-order system are  Observing the cubic polynomial Equation (28), there are two forms of the roots, namely, three real roots, a real root, and a pair of conjugate complex roots. But whatever the form of the root is, there must be a real root greater than zero. So k l is taken as the largest real root of Equation (28). According to Equations (26) and (28) the resistor. The force in the mechanical network is equivalent to current, and the velocity is equivalent to voltage. 12 According to the equivalent method, the mechanical network shown in Figure 6 is equivalent to the electrical network presented in Figure 18. Such a method will be of convenience to implement topology optimization and derivation of the pseudo-active actuator principle. According to Equations (26a)-(26e), it can be found that the expression of E E / o u is the same as Equation (12). When S 0 and S 1 are off, voltages are detected on the two sides of impedance Z o and Z g . At this time, the same as the mechanical network, the equivalent circuit also has two DOFs. Substituting Equations (29a)-(29e), the expression of E E / o u is seen to be the same as Equation (14), and E E / g u as Equation (15). The above analysis validates the equivalence of the electrical network and the proposed mechanical network.
As a briefly extended inspiration, the concept of the pseudoactive actuator, especially the concept of zero or negative dampings and the topology analysis method (i.e., mechanical-electrical analogy), might be helpful for the utilization of mechanical vibration, specifically energy harvesting, for vibration mitigation. If the proposed mechanisms of zero damping and negative damping can be added to the energy harvesters, the generated power and also efficiency will be enhanced. This is because the inherent damping of the energy harvesters might be eliminated by using negative damping. The corresponding part of the wasted power to the passive damping will be converted into generated power of the energy harvesters. On the other hand, similarly, the circuits for energy harvesting have the same issue of the inherent damping, which dissipates generated power and the circuit is not optimal. With the extension of the new concept of negative damping using the method of an equivalent electrical network, the design, and optimization of the circuits for energy harvesting, for example, matching the linear and nonlinear components of the circuits to decrease the dampings can be conducted. Combining all the above analyses, we may further realize a much higher efficiency energy harvester employing the principle of the pseudoactive actuator. A zero damping or negative damping of an energy harvesting circuit using the equivalent electrical network of the pseudo-active actuator may also be optimized and/or developed for the extremely energy scavenging. 29 7 | CONCLUSIONS Active actuators in control loops input energy to the systems, while passive/semi-active actuators only dissipate energy from the systems. This is the reason why active control systems provide better performance than passive/semi-active systems. Graphically, active actuators provide four-quadrant controllable execution capability in the force-velocity diagram, while passive/semi-active ones provide only part of the first and third quadrants. In this way, inspiration is reasonably activated: Do we have enough wisdom to use a combination of semi-active actuators to provide four-quadrant controllable execution capability? and how?
Aiming at realizing a new semi-active actuator with mechanical properties of the active actuators, this study proposed and investigated the concept and basic design principle of a pseudo-active actuator. The proposed pseudo-active actuator is composed of two semi-active actuators and one mechanical compensation mechanism. As an example, the two semi-active actuators are using a combination of semi-active controllable damping actuators and passive springs, and the mechanical compensation mechanism comprises two rack-pinion mechanisms using common gear. The mechanical compensation mechanism regulates the two semi-active actuators to move in reverse directions, and then the total output of the actuator with the mechanical compensation mechanism is the subtraction of the two semi-active actuators. The  The output force in the zero-DOF operational mode realizes the goal of the active force output by the semi-active actuator, that is, the arbitrary output force is achieved by adjusting the difference between the damping coefficients of the two semi-active actuators. Switching the connection, one DOF is added into the system and the pseudoactive actuator transits its operational mode. The possibility of using semi-active actuators to realize active control is verified from the energy point of view, and a corresponding proof-of-concept study of the pseudo-active actuator in this operational mode is also provided.
The 2DOF operational mode of the pseudo-active actuator is realized by disconnecting to the mechanical ground. It is found that compared to the conventional 2DOF model, the two masses of the pseudo-active actuator in the 2DOF operational mode are decoupled in terms of damping ratios. The stability of this operational mode is better than the conventional one under the same nondimensional parameters. As a result, the stable dimensional parameter set for the 2DOF operational mode is given. In addition, using the pseudo-active actuator, the concept of a dual-hook automobile suspension or a pseudo-active suspension is proposed. Preliminarily analysis was conducted via an application of the new 2DOF operational mode in the monocoquepowertrain system. As compared with the conventional 2DOF monocoque-powertrain system, it is found that the new 2DOF model has a smaller gain in the low-frequency range with the controller off (i.e., passive system), but the gain in the high-frequency range is greater than that of the conventional one. Finally, a mechanicalelectrical analogy work was implemented for more convenient analysis, topology optimization, and derivation of the proposed concept.
The corresponding analysis of the operational modes of the mechanical network and the equivalent electrical network is provided.
On the other hand, the combinations of the (controllable) negative stiffness actuator and (controllable) damping actuator, found in the recent literature, are regarded as a specific case of the pseudo-active actuator.
The (controllable) negative stiffness provides the capacity for translating the first and third quadrants output to second and fourth quadrants, and the four-quadrant controllable execution capability is thus realized. Experimental validations on the significant improvement of the control systems support all the anticipated advantages of the pseudo-active actuators.
As a result, the proposed concept and principle of the pseudoactive actuator might be of help and significance for the wide performance-centered applications, such as vehicle suspensions and energy harvesters and their corresponding circuits, for providing an operable approach on zero-damping or even negative damping.
Thereby great improvement of the vibration attenuation or energy scavenging efficiency might be obtained.