Optimal Design of a Damped Single Degree of Freedom Platform for Vibration Suppression in Harmonically Forced Undamped Systems

In almost any machine, the periodic motion of mechanical parts results in applied harmonic forces, which lead, in most cases, to unwanted machine vibration. For lightly damped systems, the vibration amplitude, which is closely dependent on the forcing frequency, can reach large values when the latter is close to one of the system's natural frequencies. To overcome this problem, many techniques were proposed to control and attenuate this unwanted vibration. The vibration absorber is one of the most famous passive vibration suppression techniques. When attached to a system, this simple device can significantly reduce the vibration of its host if properly designed.

This clever idea was first patented by Frahm [1] a long time ago in 1911. Since then, vibration absorbers have been the subject of a large volume of publications. The first analytical treatment of the damped vibration absorber attached to an undamped primary system was conducted by Ormondroyd and Den Hartog [2]. The optimal absorber parameters were first determined by Den Hartog [3] in an analytical closed form using the invariant points method. The analysis is based on the existence of two points of the frequency response function that are independent of the absorber damping. First, the two points were adjusted to equal heights by a proper choice of the tuning ratio. Then, two suboptimal damping ratios were obtained by forcing the frequency response function to pass horizontally through each fixed point. The approximate optimal solution was determined from a convenient average of these two solutions. Brock [4] derived an analytical expression for the optimal damping ratio using a simple perturbation method. This classical absorber design theory can be found in vibration textbooks (e.g., Refs. [5,6]). Recently, the exact analytical solution of this problem was derived by Nishihara and Asami [7] by equally leveling the peaks of the frequency response function. It is shown that the approximate solution based on the invariant points method is highly accurate and closely matches the exact optimal solution. Snowdon [8] determined the optimal parameters using the approximate method of an absorber attached to its host with a rubberlike material. The invariant points method was extended to multidegree of freedom systems by Ozer and Royston [9].

The works cited thus far correspond to damped vibration absorbers attached to harmonically forced undamped primary systems. The frequency response function of a damped primary system coupled with a damped absorber does not exhibit invariant points. Therefore, the invariant points method cannot be used to determine the approximate optimal parameters in a closed form. Furthermore, the exact optimal solution cannot be obtained analytically, because the expression of the objective function becomes complicated. An exception is found in Asami and Nishihara [10], where hysteretic damping is present in both the primary system and absorber. Bapat and Kumaraswamy [11] determined the optimal parameters in closed form for the Lanchester damper case. In all other cases, the optimal parameters are calculated numerically for a range of the mass and damping ratios of the system, as in Thompson [12], Randall et al. [13], Soom and Lee [14], and Pennestri [15]. The numerical results were presented in ready-to-use graphs for the design of absorbers coupled to damped primary systems. Designs of absorbers with multidegrees of freedom have been proposed (e.g., Febbo and Vera [16] and Febbo [17]); the reader is referred to the references therein for a more exhaustive coverage of this aspect of the research.

In the classical absorber setup, which is shown in Fig. 1(a), the primary system is modeled as a mass attached to the ground by a spring and damper, and the absorber is directly coupled to the primary mass. In this setup, the absorber position on the primary system should be carefully chosen to ensure that the latter is able to withstand the absorber's weight and forces resulting from the absorber vibration. In some cases, one would fail to find a suitable attachment point for the absorber, since it results in a large concentrated load to be applied to a small area of its host structure. Delicate instruments with flimsy structures are examples of systems that cannot tolerate large loads on their external structures. For such systems, it is sometimes more suitable to have them attached to a platform, which acts as a vibration absorber. The proposed vibration absorber setup is shown in Fig. 1(b). The objective of this work is the determination of the optimal parameters of the new absorber setup for vibration reduction in harmonically forced undamped systems. In Sec. 2, the equations of motion are derived and the frequency response functions are written in terms of dimensionless parameters. The optimization strategy is detailed and discussed in Sec. 3. The optimal tuning and damping ratios are determined in Secs. 4 and 5, respectively. The results are discussed in Sec. 6, where the optimal mass ratio is calculated. Concluding remarks and directions for future works are given in Sec. 7.

A harmonically forced undamped primary system attached to a viscously damped platform is shown in Fig. 2, where M and m denote the primary system and absorber masses, respectively. K and k are the primary system and absorber stiffness constants, respectively, and c is the absorber viscous-damping constant. The equations of motion of the system take the form

The objective of this work is the determination of the optimal absorber parameters such that the steady state response of the primary system is minimized for all forcing frequencies. This is achieved by solving a minimax problem, in which the statement reads as follows: For a given mass ratio μ of the system, determine the optimal tuning ratio$fopt$and damping ratio$ζopt$such that the maximum of the objective function$X/δst$is minimized$∀g$.

This problem is solved using a method based on invariant points, which is similar to that used by Den Hartog for the classical absorber setup. In Sec. 3, the invariant points, which are independent of the absorber damping, are determined and the optimization procedure is detailed.

Let $H(g)=X/δst$ denote the objective function. Before proceeding with the solution of this problem, it is important to note special features of H(g). The plots of the objective function for $μ=3$⁠, f = 0.5, and three different values of the damping ratio ζ are illustrated in Fig. 3. The figure clearly shows the existence of three invariant points, depicted by Q₀, Q₁, and Q₂, which are independent of ζ. The first fixed point Q₀ corresponds to the static deflection of the system (i.e., for $g=g0=0$⁠). The frequency ratios g₁ and g₂ associated with Q₁ and Q₂ are given in Appendix  A.1. Since these fixed points are independent of the damping ratio ζ, it is then imperative to find them suitable positions first by carefully choosing the tuning ratio f, as this will ease the optimization problem. Then, the optimal damping ratio can be determined by forcing H(g) not to exceed the highest invariant point. Therefore, understanding the behavior of these points as f varies is essential to correctly tune the system. To achieve that, the objective function is calculated at these points for f = 0, and it is shown in Appendix  A.2 that

A sketch of the invariant points positions at f = 0 is shown in Fig. 4(a). The rate of change of H(g) with respect to f is calculated at each invariant point in Appendix  A.2, where it is shown that

The arrows in Fig. 4(a) indicate the direction of motion of these points as f increases from zero. When f = 0 and $∀μ$⁠, Q₀ starts at infinity and is lowered as f increases. When $μ>1$⁠, Q₁ starts at a position lower than that of Q₂ and higher than or equal to that of Q₂ if $μ≤1$⁠, as depicted in Fig. 4(a). As f increases, Q₁ is raised, whereas Q₂ is lowered $∀μ$⁠. This means that some trade-off relationship exists between Q₀, Q₁, and Q₂. In the case of the classical absorber, a trade-off relation exists between two invariant points, which are independent of the absorber damping, and the optimal tuning is reached by equally leveling the invariant points. In this problem, and since Q₁ always moves in a direction opposite to that of Q₀ and Q₂ as f varies, two trade-off relations exist: one between Q₁ and Q₀ and another one between Q₁ and Q₂, depending on the value of mass ratio μ. Consider the case where $μ≤1$⁠; by inspecting Fig. 4(a), Q₁ is higher than or equal to the level of Q₂, and as f varies, optimal tuning is reached when Q₀ and Q₁ meet at the same level. At this point, Q₂ is lower than the Q₁–Q₀ level and the system is properly tuned. Now, when $μ>1$⁠, two scenarios are possible. In the first, since Q₁ is lower than Q₂, as f is increased, the two points will meet at some level. If, at that state, Q₀ is lower than the Q₁–Q₂ level, the system is properly tuned. If not, tuning is achieved by further increasing f so that Q₁ and Q₀ meet at the same level. Thus, the optimal tuning ratio will be calculated from two different tuning conditions defined as follows:

1. Tuning condition 1 (⁠$TC1$⁠): adjusting Q₁ and Q₀ to the same height and ensuring that Q₂ is lower than the Q₀–Q₁ level. This scenario is depicted in Fig. 4(b).

2. Tuning condition 2 (⁠$TC2$⁠): adjusting Q₁ and Q₂ to the same height and ensuring that Q₀ is lower than the Q₁–Q₂ level. This scenario is depicted in Fig. 4(c).

The choice of the correct tuning condition depends on the mass ratio μ. After calculating the optimal tuning ratio, the optimal damping is obtained by forcing H(g) not to exceed the Q₀–Q₁ or Q₁–Q₂ level. In Sec. 4, the optimal tuning ratios are obtained in analytical closed forms in terms of μ.

It is shown in Appendix  A.5 that Eq. (9) is verified $∀μ>2$⁠. Finally, if $f=ft2$ and $μ>2$⁠, the two points Q₁ and Q₂ will be at the same level and Q₀ lower than that level, thus satisfying $TC2$⁠. This case is depicted in Fig. 4(c). The domain of definition of $TC1$ (i.e., $0<μ<2$⁠) is simply the complement of $μ>2$⁠. Similarly, when $f=ft1$ and $0<μ<2$⁠, Q₀ and Q₁ will become equally leveled and Q₂ lower than that level, as shown in Fig. 4(b). It is important to note that, when $μ=2,ft1=ft2=1/2$ and optimal tuning corresponds to all three points, namely, Q₀, Q₁, and Q₂ being at the same level. This scenario is illustrated in Fig. 4(d). The optimal tuning ratio is plotted in Fig. 5, in which the plot is split into two parts. The first corresponds to $0<μ<2$⁠, where $TC1$ holds and $ft1$ is plotted, and the second part of the plot corresponds to $μ>2$⁠, where $TC2$ holds and $ft2$ is plotted. Figure 5 clearly shows that the lowest optimal tuning ratio corresponds to $μ=2$ and is equal to $1/2$⁠. When the platform mass is equal to zero (i.e., $μ=0$⁠), the stiffness K of the primary system becomes directly connected in series with a resilient element comprised of the platform spring k and damper c. On the limit, $ft1$ tends towards infinity as μ tends towards zero. Now, when m is too large, the main mass can be assumed to be connected directly to the ground through the spring K. On the limit, $ft2$ tends to one as μ tends towards infinity.

$TC1, f=ft1$ and $0 < μ < 2$⁠. After calculating the optimal tuning ratio $ft1$ defined for $0<μ<2$ by equally leveling Q₀ and Q₁, the optimal damping ratio is determined such that the Q₀–Q₁ level is not exceeded by H(g). The system considered has two degrees of freedom, and thus, its frequency response function is expected to have at most two peaks, corresponding to the system damped resonant frequencies. Since Q₀ and Q₁ are independent of ζ, the best choice of the latter would be that resulting in H(g), with two peaks coinciding with Q₀ and Q₁. In this case, the frequency response function will not exceed the Q₀–Q₁ level $∀g$⁠, thus resulting in an exact optimal solution to the problem. To better visualize the optimal configuration of H(g), Fig. 6 shows H(g) for $μ=0.8,ft1=0.93$⁠, and three different values of ζ. The optimal configuration of H(g) corresponds to $ζ3=1.23$⁠, where Q₀ and Q₁ are its two maxima. Two other nonoptimal shapes of H(g) corresponding to $ζ1=1.3$ and $ζ2=1.1$ are shown.

The range of μ for which the inequalities in Eq. (11) are satisfied cannot be obtained analytically, because the expressions of $∂g,gH(g0)$ and $∂g,gH(g1)$ are too complicated. An alternative numerical approach is used, where $∂g,gH(g0)$ and $∂g,gH(g1)$ are plotted in Fig. 7 in the range $0<μ≤2$⁠. Figure 7(a) indicates that Q₀ is a peak for $0<μ≤0.954$ and Fig. 7(b) that Q₁ is a peak for $0<μ≤1.106$⁠. When $0.954≤μ≤1.106$⁠, Q₀ is a minimum and Q₁ a maximum. This case is illustrated in Fig. 8(a), where $μ=1.0$⁠. When $1.106≤μ<2$⁠, both Q₀ and Q₁ are minima; this case is depicted in Fig. 8(b), where $μ=1.5$⁠. Finally, it is concluded that the optimal damping ratio $ζopt1a$ yields an optimal solution for the range $0≤μ≤0.954$ only, where both Q₀ and Q₁ are maxima.

The unknowns of the two equations in Eq. (21) are the optimal damping constant $ζopt1b$ and L. A closed form solution of $ζopt1b$ is not possible in this case, and thus, the latter is calculated numerically by solving these two nonlinear equations for a given μ. The method used in the calculation of $ζopt1b$ was first presented in Ref. [7] and used for the calculation of the exact optimal parameters of the classical vibration absorber.

The optimal damping ratio is plotted in Fig. 9. The range of μ is split into three parts, where $ζopt1a,ζopt1b$⁠, and $ζopt2$ are defined and plotted. The first two parts of the curve correspond to $f=ft1$ and the third to $f=ft2$⁠. When the mass of the platform is zero, the primary system becomes directly connected to a resilient element made of the spring k and damper c of the platform. On the limit, as μ tends towards zero, $ζopt1a$ tends towards infinity. Finally, when the mass of the platform is too large, the primary system becomes as if it is directly connected to the ground, and on the limit, as μ tends towards infinity, $ζopt2$ tends towards zero.

The optimal parameters of the platform absorber are determined using a method similar to that used by Den Hartog in the determination of the optimal parameters of the classical absorber. The optimal tuning and damping ratios are determined separately. The range of the mass ratio μ is split into two parts, and an optimal tuning ratio is calculated for each range. The first tuning condition, which corresponds to $f=ft1$ (Eq. (6)), holds for $0<μ≤2$⁠, and the second tuning condition corresponds to $f=ft2$ (Eq. (8)) and holds for $μ≥2$⁠. Then, the optimal damping ratio is calculated such that the two peaks of H(g) are at the same height. For $0<μ≤0.954$⁠, a closed form expression for the optimal damping $ζopt1a$ is derived and the peaks coincided with the invariant points, yielding an exact solution to the problem. For $0.954≤μ≤2$⁠, the optimal damping $ζopt1b$ is calculated from Eq. (21). For $μ≥2$⁠, the optimal damping $ζopt2$ is obtained analytically as in Eq. (22). For $μ≥0.954$⁠, the equally leveled peaks do not coincide with the invariant points, and hence, the solution obtained is referred to as a semiexact solution. All design parameters are summarized in Table 1. The frequency response function H(g) is plotted in Fig. 10 in its optimal shape for several values of the mass ratio μ. Three figures are shown: the first two, namely Fig. 10(a) and Fig. 10(b), correspond to the first tuning condition, where $f=ft1$⁠, and the two points Q₀ and Q₁ are at the same level. In Fig. 10(a), $ζ=ζopt1a$⁠, and the two peaks are at Q₀ and Q₁, yielding an exact optimal shape of H(g), whereas, in Fig. 10(b), $ζ=ζopt1b$⁠, and the two peaks are at the same level, which is higher than the Q₀–Q₁ level, yielding a semiexact solution. Figure 10(c) illustrates several plots of H(g) for $f=ft2$ and $ζ=ζopt2$⁠. In these curves, Q₁ and Q₂ are equally leveled and the H(g) shapes are semiexact, since the two peaks are adjusted to the same height, which is higher than the Q₁–Q₂ level. The platform frequency response function is shown in Fig. 11 for all values of μ considered in the plots of Fig. 10. The figure clearly shows the existence of a fixed point at g = 1, yielding a unity magnification factor (i.e., $Y/δst=1$⁠).

In this work, the optimal tuning and damping ratios are determined for a given mass ratio μ of the system. The same optimization problem was solved for the classical absorber setup in the literature, where it was shown that the maximum amplitude of the primary system decreases with increasing absorber mass. Hence, an optimal mass ratio does not exist in the classical setup case, in which performance increases with increasing mass ratio. As for the proposed absorber setup, the shapes of the frequency response functions shown in Fig. 10 clearly indicate the existence of an optimal mass ratio $μopt$⁠. In Fig. 10(a), as μ increases from zero, the height of the equally leveled peaks of H(g) decreases. If μ is further increased from 0.954 to 2, as shown in Fig. 10(b), the height of the peaks' decreases reaches a minimum value and then increases back again. Hence, this indicates the existence of a minimum value of the peaks' height associated with the optimal mass ratio of the system $μopt$⁠. If μ is further increased beyond 2, the peaks' height will increase, as shown in Fig. 10(c). The maximum amplitude (peak's height) of the primary system is calculated and plotted in Fig. 12. For $0<μ≤0.954$⁠, the maximum amplitude is simply equal to H (0), since the peaks coincide with Q₀ and Q₁. For $μ>0.954$⁠, it is equal to $L$ and calculated from Eq. (21). The figure clearly shows the existence of a minimum that corresponds to $μopt=1.569,ft1=0.753$⁠, and $ζopt1b=0.704$⁠. The resultant lowest maximum amplitude of the primary system that can be achieved is equal to $2.237×δst$⁠. The frequency response functions of the primary system and platform associated with $μ=μopt$ are shown in Fig. 13. To achieve the optimal performance of the proposed absorber (i.e., $∥H(g)∥max=2.237$⁠, using the classical setup, the absorber mass should equal 50% of the primary system mass (i.e., $μ≈0.5$⁠, as per Ref. [7]). Typically, the classical absorber mass is a small fraction of the primary system mass, and hence, $μ≈0.5$ is too large to be considered an acceptable mass ratio for the classical setup (see Ref. [13]). Assuming that the primary system can withstand this heavy load, implementing an absorber with a mass equal to half of the primary system mass is a real hassle. Furthermore, some systems cannot even tolerate any loading on their external structures, and hence, for such systems, the proposed setup would be more appropriate than the classical one.

The optimal design of a viscously damped platform for vibration suppression in undamped single degree of freedom systems is proposed. For a given mass ratio μ of the system, the optimal tuning and damping ratios were determined with the aim of minimizing the maximum of the primary system frequency response. The solution was obtained by first setting two of three invariant points, which are independent of the damping ratio to equal heights, then by equally leveling the two peaks of the objective function. This was made possible due to the existence of a trade-off relation between the three invariant points and two peaks. Two different expressions of the optimal tuning ratio were determined analytically, each corresponding to a range of the mass ratio, where two of the three invariant points are set to equal heights. The range of μ over which the first tuning condition is defined is split into two subranges. In the first subrange, the optimal damping was determined analytically and led to the exact solution because the two peaks coincided with the equally leveled invariant points. The optimal damping corresponding to the second subrange is the solution of a nonlinear equation and led to a semiexact solution, where the two peaks do not coincide with the fixed points but are set to the same height. For the second tuning condition, a semiexact solution was obtained where the optimal damping ratio was determined analytically. It is shown that an optimal mass ratio $μopt$ exists, unlike the case of the classical setup, where the absorber performance increases with its increasing mass.