The Analysis and Control of Longitudinal Vibrations from Wave Viewpoint

Contributed by the Technical Committee on Vibration and Sound for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received July 2001; Revised April 2002. Associate Editor: B. Yang. Vibrations can be described as a linear combination of the modes of a structure 1. An alternative is to describe vibrations as propagating waves travelling in the structure. It was found that the two descriptions often give enlightening complementary perspectives 23. Wave control can be feedforward 4. It can also be feedback. There exist two types of feedback wave control, namely the noncollocated and the collocated feedback wave control. Longitudinally vibrating rod (or equivalently, transversely vibrating string) is studied in this Technical Brief. Though the classical modal approach can be used to analyze and control the vibrations in such structures 15, a spillover problem exists due to modal truncation dictated by the practical implementation of the controller. Feedforward wave cancellation was adopted 6, however, as pointed out by the authors, it is sensitive to both amplitude and phase errors. As to the feedback wave approach, noncollocated feedback wave cancellation was designed to control the vibrations in an axially moving string 78, in which the control is designed based on the global properties of the structure. The collocated feedback wave control strategy is relatively new; so far all the related researches have solely dealt with flexural motion in beams 9101112. In this Technical Brief, the analysis and control of longitudinal vibrations in a rod from feedback wave viewpoint are synthesized. Both collocated and noncollocated feedback wave control strategies are explored. The control design is based on the local properties of wave transmission and reflection in the vicinity of the control force applied area, hence there is no complex closed form solution involved. The controller is designed to achieve various goals, such as absorbing the incoming vibration energy, creating vibration free zone and eliminating standing waves in the structure. The findings appear to be very useful in practice due to the simplicity in the implementation of the controllers. It shall be mentioned that apart from the above approaches, there exist other approaches in the suppression of longitudinal vibrations in a rod or transverse vibrations in a string. For example, in the past, boundary control has received quite a lot of attention 131415. The research involves either direct boundary wave cancellation or the application of the Lyapunov method. Passive approaches were also studied by researchers, such as using a stationary mass-spring-damper constraint 16 and attaching a viscous damper at the end of a structure 17 to suppress the vibrations. The equation of motion of a longitudinally vibrating uniform rod subjected to an external force px,t is described by 18(1)ρS ∂2ux,t∂t2−ES ∂2ux,t∂x2=px,twhere ρ is the mass density, E the modulus of elasticity and S the cross sectional area of the rod. u is the displacement along the rod, which is a function of the position x and the time t. If one assumes a time dependence of the form eiωt, then in the absence of the external forces, the solution of the homogeneous Eq. (1) can be written as the sum of two longitudinal wave components with wave amplitudes a+ and a− travelling in the positive and negative directions respectively, which gives (2)ux=a+e−ikx+a−eikxHere the time dependence eiωt is suppressed. k=ρ/Eω is the wavenumber. In the absence of discontinuities, the wave amplitudes along a uniform length of rod are related by (3)a+x0+x=fa+x0,a−x0=fa−x0+x,where f=e−ikx is the propagation coefficient. Incident upon a discontinuity, a propagating wave will be partially transmitted and partially reflected: (4)b+=ta+,a−=ra+.The transmission and reflection coefficients t and r depend on the type of the discontinuity. For a given discontinuity, the relationship among the incident, reflected and transmitted waves can be found through the particular equilibrium and continuity conditions 1920. For example, the waves d1± and d2± on both sides of an applied longitudinal point force Q at x=0 (Fig. 1) are related to each other as (5)d2+−d1+=q,d1−−d2−=qwhere q=−Q/2ikES and i denotes the imaginary unit. The reflection of waves at boundaries can be obtained from the equilibrium at the boundary. The natural frequencies ωn of, for example, a fixed-free rod with length L (Fig. 2) are found through free response analysis by considering the reflections at the boundaries and the corresponding propagating coefficient f=e−ikL, and using the phase-closure principle 21: (6)ωn=Eρ k=Eρ2n−12L π,wheren=1,2,…The forced responses are obtained by considering the waves on both sides of the disturbance(s) as described by Eq. (5), the reflections at the boundaries and the corresponding propagation coefficients. For example, the forced response of the fixed-free rod with length L due to an applied point longitudinal force (Fig. 3), with the origin chosen at D, is found to be (7)u=e−ikxd1++eikxd1−,forx⩽0e−ikxd2++eikxd2−,forx⩾0where d1+=ρ1I−ρ2ρ1−11−ρ2q,d1−=I−ρ2ρ1−11−ρ2q,d2+=−I−ρ2ρ1−11−ρ1q,d2−=−ρ2I−ρ2ρ1−11−ρ1qand ρ1=f1rAf1,ρ2=f2rBf2,f1=e−ikL1 and f2=e−ikL2.rA and rB are the reflection coefficients at boundaries A and B respectively. Incident upon the “active discontinuity” created by the applied point control force, an incident wave a+ gives rise to transmitted and reflected waves b+ and a− (Fig. 4), which are related through the transmission and reflection coefficients t and r as described in Eq. (4). The wave control force can be written in the frequency domain as (8)Fc=−Hωuwhere Hω is the frequency response of the controller. From the continuity and equilibrium at the force-applied point, t and r are found as (9)t=11−iH¯ωandr=iH¯ω1−iH¯ωhere H¯ω=Hω/2kES.From Eq. (9), it is clear that the controller can be designed to control the transmission and/or reflection of the incoming vibration waves. Though the controller can be of any form, velocity feedback wave control is considered below. In which H¯ω has the form (10)H¯ω=iCwhere C is the constant to be determined. It follows that the control forces in the frequency and time domain are given by the following equations respectively (11)Fc=C−2ρESiωu(12)fc=C−2ρESu˙The velocity feedback controller is designed to optimally absorb the incoming vibration energy. A control gain C is to be found to minimize both the transmitted and reflected vibration energies, that is, (13)Er=|t|2+|r|2It is found that C=1 and the minimum value of Er is 0.5. This means half of the incoming vibration energy will be absorbed by the controller. The corresponding control force is (14)fc=−2ρESu˙Figure 5 shows the frequency response (deflection per unit force) of a fixed-free rod before and after control. For convenience, both ρ and E are assumed to be 1. The length and the radius of the rod are chosen to be 0.6m and 0.1m respectively. The disturbance and the control forces are applied at 0.26m and 0.35m from the fixed end respectively. The response shown is the response at 0.55m from the fixed end. It can be seen that after control, the resonances are well damped, with the damping factors varying from mode to mode. To eliminate the standing waves, a velocity feedback wave control force is applied at one end of a rod. From the equilibrium condition at the control force applied end (which is chosen as the origin), r is found as (15)r=1−2C1+2CIt is clear that when choosing C=0.5, there is no reflection from the boundary, hence no standing wave in the rod. The control force in this case is (16)fc=−ρESu˙It shall be pointed out that the same mathematical relation was also obtained through closed form solutions from modal 17 and wave 13 standpoint. Figure 6 shows the responses (still at 0.55m from the fixed end) before and after control with such a control force applied at the free end of the fixed-free example rod. The fact indicates that the active controllers given by Eqs. (16) and (14) can be implemented passively using a viscous damper with the corresponding damping constant chosen as ρES and 2ρES accordingly. Assume the measurement sensor and the control actuator are applied at point A and D respectively (Fig. 7(a)), which are located distance L away. In general, the measurement sensor picks up responses from three sources: the input disturbance waves a+ and d−, and the upstream feedback wave dQ− generated by the control force (Fig. 7(b)), where (17)dQ+=dQ−=GQQand GQ=−1/2ikES.The relationship is best described using block diagram as shown in Fig. 8. Since the controller G forms the excitation Q from the measurement m, we have (18)Q=GmAssume that the disturbances are from upstream, that is, d−=0, we have (19)d+=fad+GGQ1−GGQfada+where fad=e−ikL. To achieve full cancellation downstream, it requires that (20)G=2ikES e−ikL1−e−2ikLIn terms of a+, the control forces are given as (21)Q=2ikESe−ikLa+=iω2ρESe−iωρ/ELa+In time domain, it corresponds to a proportional velocity feedback with a constant gain 2ρES and a time delay ρ/EL. Similar non-collocated scheme was used in the suppression of flexural vibrations in beams 22. In this Tech Brief, both collocated and noncollocated feedback wave controllers have been designed from wave viewpoint to suppress longitudinal vibrations in a rod. It was found that the controller could be designed to achieve various goals. These include absorbing the incoming vibration energy, terminating the wave transmission so as to create a vibration free zone and stopping the wave reflection at boundary so as to eliminate the standing waves in the rod. The implementations of the controllers are simple, since they are all feedback with constant gains. Furthermore, the designed active controller can easily be implemented passively. Hence the strategies introduced are very useful in practice.