Critical damping is represented by Curve A in Figure 3. With less-than critical damping, the system will return to equilibrium faster but will overshoot and cross over one or more times. Such a system is underdamped ; its displacement is represented by the curve in Figure 2. Curve B in Figure 3 represents an overdamped system. As with critical damping, it too may overshoot the equilibrium position, but will reach equilibrium over a longer period of time.
Figure 3. Displacement versus time for a critically damped harmonic oscillator A and an overdamped harmonic oscillator B. Critical damping is often desired, because such a system returns to equilibrium rapidly and remains at equilibrium as well. In addition, a constant force applied to a critically damped system moves the system to a new equilibrium position in the shortest time possible without overshooting or oscillating about the new position.
For example, when you stand on bathroom scales that have a needle gauge, the needle moves to its equilibrium position without oscillating. It would be quite inconvenient if the needle oscillated about the new equilibrium position for a long time before settling.
Damping forces can vary greatly in character. Friction, for example, is sometimes independent of velocity as assumed in most places in this text. But many damping forces depend on velocity—sometimes in complex ways, sometimes simply being proportional to velocity. Damping oscillatory motion is important in many systems, and the ability to control the damping is even more so.
This is generally attained using non-conservative forces such as the friction between surfaces, and viscosity for objects moving through fluids.
The following example considers friction. Suppose a 0. Figure 4. The transformation of energy in simple harmonic motion is illustrated for an object attached to a spring on a frictionless surface. This problem requires you to integrate your knowledge of various concepts regarding waves, oscillations, and damping.
To solve an integrated concept problem, you must first identify the physical principles involved. Part 1 is about the frictional force. Part 2 requires an understanding of work and conservation of energy, as well as some understanding of horizontal oscillatory systems. Now that we have identified the principles we must apply in order to solve the problems, we need to identify the knowns and unknowns for each part of the question, as well as the quantity that is constant in Part 1 and Part 2 of the question.
Identify the known values. So what happened to the other solution? We can get a clue by examining the two exponentially falling solutions for the overdamped case as we approach critical damping:.
As usual, we can always multiply a solution of a linear differential equation by a constant and still have a solution, so we write our new solution as.
A shock absorber is basically a damped spring oscillator, the damping is from a piston moving in a cylinder filled with oil. The opposite case , overdamping , looks like this:. To avoid the oscillation, the characteristic roots must lie in the negative real axis.
Assume that where is an integer, so and are obtained, then 5 can be simplified as. The establishing condition for 7 is , which means that. Therefore, it can be obtained that where is an integer. As a result, we have. We find that the set of is dense, but the probability density of any locating in this domain is small, so the existence condition of critical damping is strict.
From 6 , a negative damping coefficient is obtained when , which represents an energy input to the system. In this case, the system oscillation is strengthened, and there is no critical damping, while it is the opposite when ; that is, is odd, so substituting into 6 and then 10 is obtained. In summary, in 9 is an integer, is odd, and. The existence conditions of critical damping in the vibration systems with fractional derivative damping and its calculation formula are presented.
For linear 1 DOF fractionally damped systems, only when 9 is satisfied by the order of fractional operator, there is a critical value of damping coefficients. To make the solutions of 1 be without oscillation, the relation between the damping coefficient and the order is where. The curves that represent the relation between the variables in 10 are plotted in Figure 1. Take , for example, the lowest point of the curve represents the critical damping point and its corresponding damping coefficient is the critical value of damping coefficient.
It is worth noting that many previous researches on 1 DOF fractionally damped systems focus on the solutions of the characteristic equations. From this perspective, we find when , the characteristic equations only have complex or conjugate roots and they have negative real roots when. Therefore, when , it represents the overdamping coefficient, and when , it is the underdamping coefficient.
In the case of critical damping, the characteristic equation has the root , which represents the convergence rate. When increases from 0 to 2, the critical damping point is shifted to the lower right in the figure, which indicates that with larger , it turns out a smaller and larger ; that is, with a smaller eigenvalue, the system is a faster convergent.
By the analysis of the solutions of 1 , it is concluded that there is no critical value of damping coefficient, which is not against the conclusions of this paper because is not located in the set represented by 9. In fact, it is easy to understand that by reduction to absurdity, that is, when the roots s are negative real, they do not hold by substituting into 2. This means that when , the eigenvalues cannot be negative real and always contain an imaginary part.
Furthermore, we find that when , the critical damping coefficients , , and are obtained, which are consistent with the critical damping in an integer order system.
Because it is not our main objective to solve the equation and the critical damping coefficients can be obtained without analyzing the solutions, we will not return to these questions here and refer the interested reader to [ 17 , 18 ]. As is shown in Figure 2 , when , the critical damping coefficient is figured out according to the above analysis.
When , the fractional damping plays not only the role of a conventional damping, but also the role of a supplementary spring [ 19 ]. If or , the damping effect of the system will be weakened, and there is a typical behavior of the oscillation. Furthermore, the fractional order systems are easily affected by the initial state. Therefore, in practice, should lie within the range of engineering interest. Figure 3 shows the curves of decaying free motions of critical damping systems with different orders under the initial state ,.
It shows that in the case of the same other parameters, the systems with a large return back to equilibrium position faster. When , the systems are relatively slow as they go back to balance position and do not cross it. Otherwise when , the systems are relatively fast and cross through the static equilibrium position once overshoot occurs , which is different from the ordinary critical damping. Although the systems with a large return back to the equilibrium position at a faster speed, it is easily to be aroused by external excitation such as step input; the response curves are shown in Figure 4.
It is expected that under the premise of nonoscillatory, the system is not easy to be aroused by external excitation and can return back to the equilibrium position as quickly as possible when there is no external force. A switch control law is designed to make the displacement as small as possible when the system is away from the equilibrium position and to limit the time it takes to reach the asymptotically stable position when there is no external force.
The designed control law is where is the control force, and are the orders of fractional derivative, and and are the corresponding fractional derivative critical damping coefficients, is the displacement. The effectiveness of the proposed control strategy is tested by a pulse excitation.
Figure 5 shows that, under impulse input, the switching control law makes the vibration performance of the fractional order system better than that of the integer order one. According to vehicle dynamics theory, the dynamic model of the vehicle with seven DOFs is established. The seven DOFs , , , , , , and are the heave, pitch, roll displacement of the body, and the four wheels displacement, respectively. This model is similar to those used by [ 20 , 21 ], here the matrix differential equation of the model can be described as where is a vector consisting of , , , , , , and.
Equation 12 represents a passive suspension when is a zero vector. According to the linear vibration theory, the decoupled suspension system turns into isolated linear subsystems that can be controlled independently [ 22 ]. Therefore, a systematic modal decoupling method [ 23 ] is considered, with which the mass and stiffness matrix can be completely decoupled; however, the damping matrix cannot be completely decoupled generally.
Here only the diagonal elements of the damping matrix are controlled to verify the effectiveness of the control strategy. The matrix differential equation of the fully decoupled system is considered where is the vector of principal coordinates, , is the feature matrix, and is the diagonal matrix whose diagonal elements are equal to those in vector.
In 13 , , , and are seven-order diagonal matrices and, assuming that is also a seven-order diagonal matrices, seven differential equations of independent scalar function are obtained; fractional skyhook control is used here to depress each independent modal vibration.
The free vibration equations of the modal systems are considered, namely, where the control force is used to keep the system in the case of critical damping. According to the method in Section 2 , the relation between the damping coefficient and the order is obtained. When , the fractional derivative skyhook damping coefficient is equal to the fractional derivative critical damping coefficient. In the same way, it is hoped that with the fractional damping force, the modal system is not easily aroused by external force and returns to equilibrium position as fast as possible without oscillating when there is no force.
Here, a switching control law is given as follows:. In practice, with a larger or smaller , these problems, such as the limitation of actuator force and the work efficiency of the actuator, arise. In order to achieve a relatively good control effect, only the limitation of actuator force is considered. Seven skyhook damping coefficients of the system are obtained.
Equation 19 represents the force of integer order skyhook damping control strategy when. The generalized inverse matrix of is used here because it is not a square matrix. A four-wheels-correlated random road time domain model [ 24 ] is used here and the road profile is C grade. To verify the characteristics of fractional critical damping, a work condition is designed as follows: when the simulation goes to , on the left side of the vehicle, the front and rear wheels have been raised successively by road bump shaped like a sine wave with a height of 0.
Vehicle suspension parameters are shown in Notations. For validating the superiority of the fractional derivative critical damping, meanwhile avoiding the following negative effects with a large or small , in the switching control law, the orders are chosen as and. Figures 6 and 7 show that the proposed vehicle skyhook control strategy can effectively suppress the vibration of the body; both vibration amplitude and acceleration are decreased significantly; the performance especially is good after crossing the road bump.
Figure 6 shows that the vibration with fractional derivative critical damping has a better performance on amplitude responses than that with integer one. And Figure 7 shows that fractional order skyhook damping control strategy has no significant deterioration in acceleration response. But for a large or small , the acceleration responses become worse than those in integer order control strategy, and that is why the order should locate within a reasonable domain in engineering application.
Compared with many other full-car suspension control strategies, there are two main advantages for the method in this paper. Firstly, the proposed method is much more simple than most of the control methods. For example, these methods presented in [ 25 ] are also tested by a road bump and can improve the vibration performance of the vehicle, but they are too complicated.
Actually, the skyhook control strategy is one of several simple and practical methods which is widely applied. Among the full-car skyhook control algorithms, a skyhook-based asynchronous semiactive controller proposed by Zhang et al.
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