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The damping performance of a single particle impact damper
Abstract
This paper presents results from computer simulations used to investigate the damping performance of a single particle vertical impact damper over a wide range of excitation frequencies and amplitudes, particle-to-structure mass ratios, lid clearance ratios, structural damping ratios, and coefficients of restitution. Measurements of the damping performance, particle flight times, and structure contact times are presented. Performance at both the structure's undamped natural frequency and off-resonant conditions are studied in depth. Maximum damping at a fixed oscillation frequency occurs at an optimal lid height that increases with increasing mass ratio, increasing structural damping ratio, but decreases with coefficient of restitution. The corresponding maximum degree of damping increases with increasing mass ratio and coefficient of restitution, but decreases with increasing structural damping ratio. Field plots of the damping ratio are also presented as functions of oscillation amplitude and frequency to demonstrate the damper performance over a range of design parameters and operating conditions.
... When the primary structure vibrates, its kinetic energy is transferred to the masses, which then impact each other and the container. Therefore, the kinetic energy of the primary structure is decreased [8]. The auxiliary masses decrease the vibrations of the primary system by the impacts, which is why the damper is called the impact dampers. ...
... x0M , V0M , x0 and V0 are the location and the velocity of the primary and auxiliary systems immediately after each impact, respectively. When the auxiliary system impacts with the left or right walls, the required conditions are given by Eq. (8) and Eq. (9), respectively. ...
... Then the collision condition is evaluated at each instance using Eqs. (8) and (9). If there is a collision, by using Eq. ...
... The operating principle of semi-active particle damping equipment lies in controlling the friction and collision motion between particles, as well as between particles and the container. This is achieved by regulating the input conditions such as the particle filling ratio or currents [14,15]. Dynamic characteristics of such equipment are inherently nonlinear and encompass multi-body dynamic issues. ...
... Next, it is of possibility to further simplify the dynamic equation when consider the fact that the modal shape function of a slender cantilever beam abide by the orthogonality principle and modal mass normalization principle, as is expressed in Eq. (15) and Eq. (16), respectively. ...
... Finally, multiply φ j (x) at both sides of the Eq.(12) and then integrate it from 0 to l. Applying the transformation rules given by Eq. (15), (16) and (17) in the following step. After that, the dynamic equation described in Eq.(1) will be transformed into a dimensionless form: ...
Semi-active particle damping equipment have a well adaptability when it is deployed on flexible structures, but the dynamic characteristics is complicated since a large amount of particles are filled into the damping equipment and the magnetic field inner the equipment should also be considered. The fractional-order model of this kind of equipment is established, where the designed parameters kp and α are respectively associated with the filling ratio and input current. Depending on the Hertz-Mindlin contact model, the multi-body dynamic issues inner the damping equipment is solve by discrete element method, and then the fractional-order parameters kp and α are identified according to the energy distribution within a single vibration period. The numerical simulations calculated via the fractional-order model are basically consistent with the results extracted from experiments, which can validate the accuracy and feasibility of the fractional-order model. Finally, the optimized value of the input current, and the motion trajectory of the damping equipment under different scenarios are studied based on the fractional-order model. The results indicate that the optimized input current is approximately 2.5A; and with the increase of parameter kp and α, the motion trajectory will rotate in a counterclockwise direction, and the outermost curve will gradually move away with respect to the origin position.
... More recently, Duncan et al. 20 accomplished a numerical investigation of the performance of the damping of one SPID of vertical vibrations ( Figure 5) for a vast interval of frequencies, amplitudes of vibrations, damping rates, and other parameters. According to Bapat and Sankar, 21 later confirmed by Popplewell and Liao, 22 the efficiency of an ID decreases with the increase of the damping rate of a given structure, and it reaches the maximum when the rate between the frequency of loading and the natural frequency of vibration of the structure approaches unity, that is in resonance. ...
... One can note that in all the problems solved until now, the augmented Lagrangian method program converged with eight iterations, which is the minimum number of iterations adopted in the implementation. Now, trying to put more emphasis in the stationary regime, the solution of the optimization problem considering the objective function by equation (23) and constraints functions given by equations (20) to (22) will be presented. Adopting the initial design f6.0 cm; 0.200 kg; 8 mmg, with eight iterations, an optimum design with d ¼ 3.3 cm, m 3 ¼ 0.313 kg, and max(q 1 ) ¼ 8.0 mm was obtained, as shown in Figure 14. ...
... Continuing with the strategy of giving more emphasis in the stationary regime, the solution of the optimization problem given by the objective function, equation (24), and constraint functions given by equations (20) and (21) will be presented. Note that the number of design variables is now equal to 2 and there are only two constraints applied to the problem. ...
It is intended, in this work, to present some research results on the optimization of an impact damper for a structural system excited by a non-ideal power source. In the model, the impact vibration absorber is, basically, a small free mass inside a box carved in the structure that undergoes undamped linear motions colliding against the walls of the box. Whenever the mass shocks against the walls of the box, an exchange of kinetic energy between the mass and the structure may be used to control the amplitude of the dynamic response of the structure. In this work, the structure is excited by a non-ideal power source, a DC electric motor installed on it, which may present the Sommerfeld effect. A non-ideal power source is one that interacts with the motion of the structure as opposed to an ideal source whose amplitude and frequency are fixed, independent of the displacements of the structure. Here, the dynamic response of the system is computed using step-by-step numerical integration of the equations of motion derived via a Lagrangian formulation. The optimization problem is defined considering as the objective function the maximum amplitude of the structure displacement, while the design variables are the weight of the free mass and the width of the carved box. Using the augmented Lagrangian method, several optimization problems are formulated, and, solving them, the best design to maximize the efficiency of the impact damper is obtained.
... In the past, the damping effect resulting from particles was first corroborated by experimentation [7-9, 13, 18-20], but how particles dissipate the system energy was still a puzzle. To understand the nonlinear behavior of particle dampers and the mechanism of vibration reduction for the particle damping technology, the discrete element method (DEM) [21], a well-established method for modeling granular assemblies, can offer a deeper comprehension of particle damping [22][23][24][25][26][27][28][29]. However, the above investigations [22][23][24][25][26][27][28][29] mainly describe the dynamics of the damping particles, and did not consider the interaction between mechanical elements and damping particles. ...
... To understand the nonlinear behavior of particle dampers and the mechanism of vibration reduction for the particle damping technology, the discrete element method (DEM) [21], a well-established method for modeling granular assemblies, can offer a deeper comprehension of particle damping [22][23][24][25][26][27][28][29]. However, the above investigations [22][23][24][25][26][27][28][29] mainly describe the dynamics of the damping particles, and did not consider the interaction between mechanical elements and damping particles. This is because the DEM assumes contacting surfaces to be rigid bodies with no mass, and the dynamic inertia properties of the mechanical elements cannot be modeled. ...
Adding particles to mechanical elements can reduce their vibrations. Both the particles and the mechanical elements interrelate in a highly complex manner, thereby influencing the energy dissipation of the mechanical elements. The particle damping is extremely nonlinear, and the energy dissipation mechanism in such a granule–structure interaction system has scarcely been examined. This study aims to investigate the dynamic behavior and energy dissipation mechanism for a mass–spring–damper–slider system with a particle damper. A simple but robust energy method was first proposed to explore the energy dissipation mechanism, and a two-way coupled model of the discrete element method (DEM) and multi-body dynamics (MBD) was employed to analyze the complex interaction system. Three numerical benchmark tests and free vibration experiments for the system with a particle damper were conducted to validate the proposed energy method and the adopted coupled DEM–MBD model. Results show that the coupled DEM–MBD simulations reasonably agree with the corresponding experiments. The validated coupled model was subsequently employed to calculate the distribution of system energy, and to explore the effect of contact properties on the energy dissipation of the system during the free vibration process. In the mass–spring–damper–slider system with a particle damper, the damping effect resulting from particles is essentially caused by the contact forces generated when the particles make contact with the hollow box. The induced contact forces act as resistance forces to the hollow box, always do negative work, and suppress the motion of the hollow box. The energy loss of the particles primarily occurs through contact friction and contact damping when the particles are hit by the hollow box. Contact properties, such as friction and restitution coefficients, exhibit a negligible effect on the dynamic behavior of the hollow box, but substantially affect the distribution of energy dissipation in the particular system.
... Due to the simplicity of their construction, impact dampers have been widely used for structural damping applications in skyscrapers, machine tools and other lightly damped structures. Impact dampers have also been considered for use in harsh environments, such as turbo machinery blade as their effectiveness is independent of the environment [24]. The limitations of this damper is large accelerations are imparted to the structure at the time of impact, which may be undesirable. ...
... Despite their simple design, the dynamics of impact dampers can be very complex. The particle and container trajectories are continuous, but not smooth due to impacts [24]. ...
... The reduction of vibrations of the cantilever beam was found to be greatly related to the damper position and the particle-filling ratio, and is mainly attributed to the collisions between the particles. Alternatively, the discrete element method (DEM) [14] , a numerical method for modeling the motion of particles, can provide a deeper understanding of particle damping [15][16][17][18] . Chen et al. [15] used the DEM to simulate the vibration behavior of a cylindrical rod with a particle damper. ...
... Their results showed that the lower the filling ratio of the particle damper, the better the vibration reduction for the cantilever beam. Duncan et al. [17] simulated the dynamic behavior of a plate with a particle damper containing only one particle by using the DEM, reporting that the maximum degree of damping increases with the particle-structure mass ratio and the restitution coefficient. Machado et al. [18] adopted the DEM to predict the mechanical response of a bearing with a rigid or elastic housing. ...
Damping particles can be used to attenuate vibrations in mechanisms. However, damping particles and mechanical parts interact in an extremely complex manner, which affects the energy dissipation of the mechanisms. This study proposes two-way coupled models based on Multi-Body Dynamics (MBD) and Discrete Element Method (DEM) to solve granule-structure interaction problems, and uses two sets of experiments to validate the numerical model. Subsequently, the validated coupled MBD–DEM model was used to further investigate the effects of cavity size and chamber number of particle dampers on the dynamic characteristics of mechanisms. Results show that the coupled MBD–DEM simulations reasonably agree with the corresponding experiments. In the mechanism with a particle damper, under the same mass but with different cavity sizes, the effect of vibration reduction follows the sequence: 1/4 box>1/8 box>1/16 box. Under the same mass but with different chamber numbers, the degree of damping follows the sequence: single-chamber box>double-chamber box>triple-chamber box. Adding damping particles to mechanisms does not affect the vibration period, but does reduce the acceleration amplitude.
... An impact damper is a small loose mass within a main mass which can suppress undesired vibrations of the main mass. There are many advantages in using impact dampers over traditional passive devices: They are inexpensive, simple in design, robust and effective in harsh environments with a wide range of frequencies [1][2]. Applications of the impact dampers to reduce undesired vibrations of turbine blades, machine tools have been investigated [3][4][5][6][7]. ...
... Based on the Hertzian model, the following relationship holds between the Hertzian contact force (F c ) and the relative displacement of the impact mass and barrier [1][2]: ...
In the present study, free vibration of a vibratory system equipped with an impact damper, which incorporates the Hertzian contact theory, is investigated. A nonlinear model of an impact damper is constructed using spring, mass and viscous damper. As a practical case study, a five-story building is simulated in SimMechanics toolbox of MATLAB software. Performance of the impact dampers to suppress structural vibrations of the structure is studied. It is shown the impact dampers can effectively suppress undesired vibrations of all of the building floors. To show the ability of impact dampers, for quenching vibrations, the so-called "effectiveness" is introduced. Finally, it is shown the impact damper can increase the effectiveness up to 34 percents.
... Therefore, the solution of applying PIDs to dissipate vibration provides more possibilities for engineers, as there are plenty of design parameters available for a possible implementation. Actually, numerous studies have carried out extensive parametric studies to determine the optimal parameters by which the maximum damping effectiveness can be acquired (see Refs. [157,158,180,235]). ...
... Although, compared with traditional passive dampers, the particle impact damper has many attractive strengths, such as maintenance free operation, harsh environment application, and wide frequency band of vibration reduction, both the impact and particle dampers have certain shortcomings. For example, the impact damper presents positive damping performance for all the excitation amplitudes only under the condition that the excitation frequencies are at or just above the undamped nature frequency of the primary structure [235]. Analogously, the maximum efficiency of the particle damper can be achieved only when the excitation frequencies are near the resonant frequency of the primary structure [196]. ...
Structural vibration is a common phenomenon existing in various engineering fields such as machinery, aerospace, and civil engineering. It should be noted that the effective suppression of structural vibration is conducive to enhancing machine performance, prolonging the service life of devices, and promoting the safety and comfort of structures. Conventional linear energy dissipative devices (linear dampers) are largely restricted for wider application owing to their low performance under certain conditions, such as the detuning effect of tuned mass dampers subjected to nonstationary excitations and the excessively large forces generated in linear viscous dampers at high velocities. Recently, nonlinear energy dissipative devices (nonlinear dampers) with broadband response and high robustness are being increasingly used in practical engineering. At the present stage, nonlinear dampers can be classified into three groups, namely nonlinear stiffness dampers, nonlinear-stiffness nonlinear-damping dampers, and nonlinear damping dampers. Corresponding to each nonlinear group, three types of nonlinear dampers that are widely utilized in practical engineering are reviewed in this paper: the nonlinear energy sink (NES), particle impact damper (PID), and nonlinear viscous damper (NVD), respectively. The basic concepts, research status, engineering applications, and design approaches of these three types of nonlinear dampers are summarized. A comparison between their advantages and disadvantages in practical engineering applications is also conducted, to provide a reference source for practical applications and new research.
... An impact damper is a device which controls the response of the primary system through the impact between free mass (impactor) and primary system during vibration. Due to its advantages such as effective damping effect, simple structure, low cost, easy implementation, no power requirement, and suitable for harsh environment, there have been many studies on the device recently, including single impact dampers, [1][2][3][4] BBD impact dampers, 5 multi-body impact dampers, 6 particle dampers, 7-12 non-obstructive particle dampers, 13,14 and impact damper with fine particles. [15][16][17][18][19][20] While various types of impact dampers exist, they have the same underlying damping mechanism: during vibration, momentums are exchanged and mechanical energy is dissipated via collisions. ...
... Among the devices, the single impact dampers are most widely studied, and many theoretical models were developed based on the single impact dampers. 3,6,17 The studies found that the current devices typically have two symmetric impacts per cycle to achieve good damping performance. 1,2 However, the efficient damping effect can only be achieved near the resonance point, and the location far away from the resonance point has poor damping or even amplifies the vibration. ...
Impact dampers are able to reduce vibration effectively through impacts and thus have wide applications in mechanical control. This work investigated the chattering behavior of a two-degree-of-freedom impact damper. Chattering is a phenomenon of continuous collisions within a short time. A theoretical model was developed to obtain the completing position and time of chattering. The fourth-order Runge–Kutta method was adopted to numerically solve the model. The results of bifurcation diagram and temporal map were obtained with different parameters, confirming the validity of the model. Furthermore, the model showed that viscosity is an important parameter separating the completing chattering from the non-completing chattering. The results demonstrated that the model can help solve chattering of impact damper by developing better understanding of the underlying mechanisms.
... The particle damper (PD) can effectively reduce vibration by harnessing momentum transfer and energy dissipation, mainly through a combination of particle-particle collisions, interactions between particles and container [5]. Due to its impressive attributes, which include a wide frequency range for vibration reduction, robustness, reliability, insensitivity to temperature variations, and minimal impact on the underlying structure, PD has gained extensive attention from scholars [6][7][8][9][10]. In the previous study, we proposed a particle tuned mass damper (PTMD) [11] to realize a wide-frequency damping band and high durability, where the cavity of the damping system can be tuned to the main structure's natural frequency. ...
An enhanced particle inerter device (EPID) is innovatively developed to mitigate the dynamic responses of multi-story structures subjected to seismic excitations. The device dissipates energy through tuning, particle-cavity collisions, and the inertia amplification effect of the inerter. Multi-story response can be mitigated when the inerter unit connect the particle device and the cross-story. In this paper, a shaking table test was conducted to investigate the working mechanism of the EPID on the dynamic response of multi-story structures. A series of parametric studies were undertaken to analyze both the individual and combined effects of the particle damping unit and inerter unit within the device. Furthermore, a parameter optimization analysis guided by a genetic algorithm was carried out, resulting in a significant enhancement of 27.92% in displacement control and 24.41% in acceleration control when compared to the traditional design. Moreover, the energy dissipated by the main structure with the optimized EPID shows a decrease of 45.4% when compared to the traditional design.
... Results showed that the effective reduction of the vibration response is dependent not on the number of impacts but primarily on the type of collision that the impact mass collides with the main mass face-to-face. Duncan et al. [5] studied the performance of a single particle vertical impact damper and found that an impact damper designed to produce the greatest degree of damping should have a large mass ratio, a large coefficient of restitution, and should have a lid height optimized for the expected forcing oscillation amplitude. ...
Vibration plays a vital role in many structural elements. Excessive vibration can cause failure due to resonance and fatigue, malfunctioning of critical components, etc. Vibration can be controlled using passive, active and hybrid control systems. This paper presents the experimental studies carried out on the behaviour of impact dampers, which belong to the category of passive vibration control devices that are used to attenuate vibration of discrete and continuous systems. Impact damping is a technique for improving damping of a dynamic system by means of dissipation of energy due to collision of a free mass on the main mass. Experimental studies are carried out for forced vibration conditions by giving sinusoidal varying forcing function to the cantilever beam system using an exciter system and the time/frequency response function is determined. Impact mass is placed at free end of cantilever beam and the system is given constant base excitation and the damping of the system is studied by varying the frequency. Experiments are conducted for various mass ratios with different impact mass materials and the results are compared with theoretical predictions.
... Yiqing et al [30] presented a study for the optimal parameters of a single-mass impact damper embedded on a cantilever beam considering its velocity response, and it was concluded that reverse-direction collision produces higher damping than co-directional collision. Duncan et al [31] presented a numerical analysis of the damping performance of a single-single-mass impact damper in the vertical direction. It was concluded that the clearance magnitude is cruicial prameter and the optimal clearance increases with the increase in mass ratio and main structure's damping ratiobut decreases with coefficeint of restitution. ...
This research article presents a numerical approach to establish an optimal design methodology for a single-mass impact damper (SMID), which is a passive energy dissipation device with robust performance. Due to the nonlinear characteristics of SMID and a lack of analytical models, designing a single-mass impact damper with optimal combination of the parameters has been challenging. Furthermore, an uncontrolled mass of the SMID on a vibrating structure may lead to chaotic vibration responses. This study identifies a range of design parameters of the SMID to ensure non-chaotic responses and validates the optimal design combinations using an experimental prototype. The results show that a single-mass impact damper designed with the optimal combination of design parameters can provide better vibration damping and relatively steady response. This study also compares the performance of an optimized single-mass impact damper with an optimized tuned mass damper and finds that the single-mass impact damper can work more effectively than the tuned mass damper in damping free vibrations of a single-degree-of-freedom primary structure. Although the SMID cannot suppress forced vibration amplitude as effectively as a tuned mass damper (TMD) at resonance, it has the advantages of lower cost and easier installation than the TMD. Overall, this study provides a basis for the optimal design of a single-mass impact damper and resolves the issues related to design methodology and chaotic vibration response with a single-mass impact damper.
... Yiqing et al [30] presented a study for the optimal parameters of a single-mass impact damper embedded on a cantilever beam considering its velocity response, and it was concluded that reverse-direction collision produces higher damping than co-directional collision. Duncan et al [31] presented a numerical analysis of the damping performance of a single-single-mass impact damper in the vertical direction. It was concluded that the clearance magnitude is cruicial prameter and the optimal clearance increases with the increase in mass ratio and main structure's damping ratiobut decreases with coefficeint of restitution. ...
... This is illustrated in Fig. 1a. The impact damper relies on material dissipation in the contact region [5,6,7,8,9,10,11]. This is typically associated with inelastic deformation and damage. ...
The Vibro-Impact Nonlinear Energy Sink (or impact damper) is well-known for its ability to engage into transient resonance captures with arbitrary frequencies and thus has inherent broad-band effectiveness. Its working principle relies on (recurrent) energy localization and local dissipation within the contact region. Dissipative (inelastic) collisions are inevitably associated with damage and challenging to predict. Recently, it has been shown theoretically that the device is effective even for purely elastic collisions when the energy is (almost) irreversibly transferred from the critical low-frequency modes to high frequencies.In that case, the device is more properly termed Impact Energy Scatterer (IES). In the present work, we experimentally validate, for the first time, the working principle of the IES. To this end, we design a test rig consisting of a cantilevered beam, hosting a spherical impactor inside a cavity at its tip. The resonant vibrations of the lowest-frequency bending mode are reduced by a factor of 10-20. Given that the IES weighs less than 1% of the host structure, this corresponds to a paramount vibration mitigation capability. We achieve excellent agreement between the measurements and the numerical predictions obtained by modeling the impacts as perfectly elastic. We also demonstrate that the dissipation in the contact region is negligible, while a substantial amount of energy is scattered to higher frequencies, validating the theoretically proposed working principle.
... The movement of the auxiliary mass inside the box leads to momentum transfer from the vibrating structure to the impact mass, which results in reducing the vibration amplitude of the main structure [17], see Fig. 1(a). In comparison to conventional passive damping, the impact damper poses several advantages, like conceptual simplicity, affordable, insensitive to extreme temperatures, and robustness towards the harsh environmental condition [18]. However, high noise levels, material deterioration, local deformation, and sensitivity towards the excitation level may restrict its application [19]. ...
A particle damper is a passive vibration control technology that utilizes the high damping properties of granular materials for reducing the vibration amplitude of a structure over a wide frequency range. Energy dissipation of particle damper is a highly nonlinear and complex physical phenomenon [1]. Previous studies have shown that the damping mechanism of a particle damper depends on several factors, like particle size and shape, granular material, filling ratio, and the number of particles [2-5]. Out of these parameters, the type of granular material used in the particle damper plays a major role in reducing the vibration amplitude of a mechanical structure. Therefore, the current contribution aims to investigate the influence of 20 different granular materials on vibration attenuation. The granular materials under investigation are subdivided into two major groups, namely: soft particles, like rubber granulate, and hard particles, like steel balls. Furthermore, this paper proposes a hybrid particle damper in which two different types of granular material mixtures are used, i.e. a particle damper in which for instance soft particles are mixed with hard particles. Moreover, in this contribution, fine particles, like rubber powder are mixed with hard granular materials like lead shot, which can be seen as an extension of the fine particle impact damper concept [6]. The humidity-dependent behavior of particle dampers is also another important issue, which is addressed in this paper. To investigate the vibration attenuation efficiency of all the granular materials and their mixtures including the humidity-dependent behavior, a laser scanning vibrometer device is used. Generally, the relationship between the vibration response and the gran-ular material mass is rarely addressed in the literature, which can restrict the industrial application of particle dampers. Hence, the additional mass of the granular material is also a focus of this paper. The experimental investigation shows that the vibration response of the test specimen is significantly lower for particles with lower densities in the low-frequency range. Furthermore, an excellent damping efficiency can be also observed for the particle damper with a granular material mixture. The results obtained in this study are not restricted to any special structure and can be implemented in several industrial applications, where vibration and noise attenuation plays a major role, like automotive, aerospace, wind turbine, medical technology, mining, etc.
... If the collision time is no longer assumed to be zero, there is a need to consider the ''sticking'' effect of the two objects during collision (Friend and Kinra 2000;Renaud et al. 2009). Though the sticking effect has formerly been considered in numerical modeling of particle dampers (Duncan et al. 2005;Ramachandran and Lesieutre 2008), it has not, to the author's knowledge, been demonstrated for vibration energy harvesters. ...
We carried out a numerical analysis and an experimental investigation of the nonlinear resonance characteristics of an electromagnetic vibration energy harvester driven by impacts with a ball. In our prototype device, a cantilever beam is used as a frequency up-converter, and a rigid spherical ball is used to generate impact forces acting upon the cantilever. The purpose of this study is to investigate the feasibility of downsizing the device to assess the potential of these devices for energy harvesting from human-induced low-frequency vibrations. To this end, we developed a mathematical model of the device, which accounts for the sticking motion after impact. The model accuracy was verified by comparing the nonlinear harmonics of the vibration amplitude with the experimental results. The numerical analysis was implemented in an in-house code in MATLAB. On impact with the ball, the power density was 0.8 µW/cm³ at 10.5 Hz for a cantilever with a natural frequency of 34 Hz, which is 140 times larger than that without the ball. It was found that with a shorter cantilever the power density achieved is comparable to larger cantilevers owing to the high-frequency nonlinear harmonic vibrations. The critical gap size between the cantilever and the ball where the power density drops off was estimated from the numerical analysis.
... However, these work generally focus on tuning aspects and on the mathematics of control theory. Other works present theory on impact dampers / pounding dampers [16,17,18,19,20,21], particle dampers [22,23]. For a more detailed discussion of impact dynamics and stability of periodic solutions, see e.g. ...
Mass dampers are widely used in engineering applications. We consider the effects of limitations on the damper amplitude. Using simple methods to analyze very general mass dampers, we find an upper limit to the damping. The maximum damping logarithmic decrement is δmax = 4μα, where μ is the mass ratio, and α isthe amplitude ratio of damper to structure amplitude. The result is further discussed in relation to Tuned Mass Dampers (TMDs), which can performvery well if there is enough avaliable space. In practice, amplitude limits always apply, and our result can be used to relate these to the damper performance.Our result also applies to active devices, which have to obey the limit mentioned above. Simulated tests of TMDs and other mass dampers are described. The damping is measured both by decay tests and by forced motion test. The methods agree well in the amplitude-limited regime. In other cases, decay tests are difficulet to interpret, indicating that one needs to be very careful whenmeasuring damping of 2DOF systems based solely on decay tests. We hope that our result may inform the selection and design of mass dampers in the future, where one should consider amplitude limits as the very first step.
... Rigid masses contribute no appreciable internal damping, but impart inertia that can prolong vibration. In contrast, particle impact dampers [8][9][10] (PIDs) hold hundreds to thousands of particles as a single concentrated mass that moves freely within a rigid shell to produce damping via particle impact, [9][10][11][12][13][14] friction, [15][16][17][18] and hysteretic particle deformation. 17,18 PIDs are effective over wide temperature and frequency ranges and can be found in applications including tennis rackets, 19 spacecraft, 20 turbomachinary, 21 and civil structures. ...
In this study, we explore the vibration damping characteristics of singular liquid drops of varying viscosity and surface tension resting on a millimetric cantilever. Cantilevers are displaced 0.6 mm at their free end, 6% their length, and allowed to vibrate freely. Such ringdown vibration causes drops to deform, or slosh, which dissipates kinetic energy via viscous dissipation within the drop and through contact line friction. Damping by drop sloshing is dependent on viscosity, surface tension, drop size, and drop location. A solid weight with the same mass as experimental drops is used to compare against the damping imposed by liquids, thereby accounting for other damping sources. Neither the most viscous nor least viscous drops studied imposed the greatest damping on cantilever motion. Instead, drops of intermediate viscosity strike the most effective balance of sloshing and internal dissipative capacity. Very thin cantilevers with sloshing drops express more than one dominant frequency and vibrate erratically, often shifting phase, presenting a challenge for quantification of damping. Finally, we introduce a new dimensionless group aimed at incorporating all salient variables of our cantilever-drop system.
... Instead of using a damping element, the impact damper realizes the exchange of energy between the structure and the free-moving mass by pounding boundaries Darby, 2006, 2009). Studies have shown that the impact damper is simple to design (Popplewell and Liao, 1991) and effective to be used in harsh environments (Duncan et al., 2005). Song et al. developed a novel pounding tuned mass damper (PTMD) (Li et al., 2015;Lin et al., 2017;Song et al., 2016;Zhang et al., 2013), which can be regarded as a combination of the traditional TMD and the impact damper. ...
In this article, a clutching inerter damper is introduced into the conventional tuned mass damper to replace the typical damping element. Regarding the limitation of the typical damping element, the reformed clutching tuned mass damper system is more flexible in parameter design than the optimal tuned mass damper, which may be constrained by the manufacturing process to realize the too small or too large damping coefficient. To investigate the effectiveness of the clutching tuned mass damper, some fundamental analyses are first conducted on the clutching tuned mass damper, and results show that the clutching tuned mass damper system can achieve a similar control effect to the optimal tuned mass damper design. Considering the inherent nonlinearity of the clutching tuned mass damper, the equivalent linearization is performed based on the equivalent linearization parameters drawn from the single-degree-of-freedom system with clutching inerter damper. The equivalent linear system of the clutching tuned mass damper system has been proved to be quite accurate to approximate the nonlinear clutching tuned mass damper system. Based on the equivalent linear system, the performance evaluation and optimal design of the clutching tuned mass damper system are carried out by numerical analysis and analytical solution. Results have shown that there is an optimum inertance for the clutching tuned mass damper to achieve the optimal performance, and the optimum inertance is related to the structural damping ratio and the tuned mass ratio. Finally, the effectiveness of the clutching tuned mass damper system and its equivalent linear system in a multi-degree-of-freedom structure is verified by a numerical study.
... The envisaged periodic structure comprises an impact damper [34,35,36] as the unit cell. Impact dampers are studied since many years and their favorable properties have been already demonstrated [37,38,39,40]. Still, their behavior in periodic configurations remains largely unexplored. ...
The emergence of metamaterials as an alternative concept for the mitigation of structural vibration is increasingly attracting the interest of scientists and engineers. Recent studies confirm that the development of periodic structures assembled on the basis of unit cells of favourable properties, results in a filtering effect, preventing the propagation of vibration lying within a specific frequency band. The, so called, meta-structures are thus characterized by a frequency bandgap. For the purposes of structural vibration attenuation, two main interrelated challenges are currently associated with the conceptual design of such structures: the first corresponds to the feasibility of reducing the lower threshold of the band gap within a practical setting, while the second pertains to increasing the breadth of the bandgap. One approach to addressing these challenges lies in the use of the so-called "rainbow traps", i.e., meta-structures consisting of unit cells with different properties. An alternative strategy pertains to use of nonlinear unit cells. Both schemes are so far little unexplored in terms of applicability for structural vibration miti-gation. This study attempts to contribute to this research path, by exploring the nonlinear approach, by assessing the properties of finite lattices with unit cells composed of impact dampers. These devices have already demonstrated their attenuation potential, yet, their investigation has been mostly limited to systems of single, or two-degrees-of freedom. To this end, a one-dimensional finite lattice is herein considered , and the analysis is conducted over critical structural parameters, including the number and the individual stiffness properties of unit cells, the mass ratios, etc. The preliminary results demonstrate the potential of meta-structures composed of properly-designed impact dampers for vibration attenuation.
... Figure 1.1 shows these varieties of the particle dampers [1]. PD, which is capable to work under rough conditions, is improved to be effectively used in the wider range of frequencies and needed less maintenance [6]. Damping properties of the particle damper were defined as a combination of friction, elastic and plastic collision losses with momentum changes. ...
Passive dampers to treat the excessive structural vibration has long been researched and used in industry. Multiple particles placed in a container can be used to dissipate excessive vibration through the inelastic collisions between particles and cavity of the damper has been shown. However, their application is usually limited to treating certain modes of structural vibration and their performance is highly dependent on location. Another limitation of the particle dampers is nonlinear character caused by discontinuity and randomness of the collisions and velocity of the particles. In this paper, it is proposed to modify a particle damper into a metamaterial type structure in order to expand the applicability range. Metamaterials are known to exhibit subwavelength performance offering superior vibro-acoustic properties over a wide range of frequencies. To maintain metamaterial properties, the casing of the particle damper is designed to resonate near selected modal frequencies. The Bloch-Floquet Theory is applied in studying the singly periodic arrangement of the resonating damper shells with and without particles. Finally, the nonlinear effects observed in the metamaterial structure made of particle dampers are modelled numerically to predict their vibro-acoustic effects in finite structures. The theoretical, numerical predictions are compared with the experimental results.
... Conventionally, DPs are filled in drilled holes (Xu et al., 2005;Xu et al., 2004) or a container filled with DPs are attached to the vibrating structures. The parameters involved in impact damping problem such as packing fraction, mass ratio, clearance, material and dimension of the DPs and enclosure, nature of vibration environment has been extensively studied (Vinayaravi et al., 2013;Masri and Caughey, 1966;Masri, 1970;Duncan et al., 2005;Gharib and Ghani, 2013 ...
Equipment panels of a spacecraft are made up of a sandwich composite with aluminum face sheets and a honeycomb (HC) core. The HC sandwich plate responds to the launch vibration loads subjecting the equipment mounted on it to a high level of accelerations at resonances owing to a lower natural damping. Damping particles (DPs) when inserted in the empty cells of a HC core improve the damping characteristics and reduce the resonance responses. In this work, we present a mathematical model governing the motion of the cell walls, DPs and HC plate under dynamic loading. The discrete element method (DEM) has been used to model the dynamics of the DPs wherein the contacts are modeled using modified nonlinear dissipative Hertz contact theory in conjunction with Coulomb friction. The effect of DPs on the responses at resonances, damping, and frequency response function (FRF) of the HC plate is obtained. Numerical and experimental studies were conducted on a HC plate where a selected portion of the plate was filled with DPs. The HC plate was subjected to sine sweep base acceleration at the edges to study the effect of DPs on the dynamic characteristic of the plate. The damping ratios and resonance peaks of the lower four modes of the HC plate, excited up to 1000 Hz, obtained experimentally from the FRF measurements and numerically from the DEM model compare well. The damping ratios, response at resonances and the FRF profiles are also similar. Significant improvement in damping ratios and attenuation of vibration level has been observed.
... In addition, Nagarajaiah (2000) proposed a semi-active variable stiffness based TMD to realize robust control performance by allowing adaption to structural frequency changes (Varadarajan and Nagarajaiah, 2004;Nagarajaiah and Sonmez, 2007). Furthermore, the impact damping mechanisms were also widely utilized for dissipating vibration energy through inelastic impacts between particles against a wall coupled to the host structure (Bapat and Sankar, 1985;Duncan et al., 2005), or through particle to particle collisions (Lu et al., 2011a(Lu et al., , 2011b(Lu et al., , 2018Zhang et al., 2018). Most recently, Song et al. (2016) integrated the TMD device with the impact damping mechanism to form a pounding tuned mass damper (PTMD) which functioned by enabling the tuned mass to pound against viscoelastic materials and thus suppress vibration motions of pipelike structures in air Zhang et al., 2015) and underwater (Jiang et al., 2017). ...
This paper explores the feasibility of leveraging the damping generated by the friction between movable flange-mounted ball bearings and a stationary shaft. This bearing–shaft assembly is integrated with a tuned mass damper to form a frictional tuned mass damper (FTMD). The friction coefficient and the equivalent viscous damping ratio of the proposed FTMD were experimentally obtained based on different cases of glass, steel, and aluminum slide shafts. The proposed FTMD was modeled and simulated numerically to study its ability to suppress vibrations on a single degree of freedom structure. Furthermore, a parallel experimental validation of the FTMD was also executed to verify simulation results. Results from both experiments and simulations demonstrated that the proposed FTMD device was able to significantly improve the damping ratio of the primary structure from 0.35% to 5.326% during free vibration, and also to suppress around 90% of uncontrolled structural response at a tuned frequency. In particular, the frequency responses, among the tested shaft materials, suggested that the selected steel slide shaft practically provided a near-optimal damping coefficient, thus the proposed FTMD was able to considerably reduce structural resonant peak amplitudes over the tested excitation frequency domain.
... The impact/pounding between the mass and the delimiters dissipates energy. The impact damper has some advantages [79] such as simple construction, easy adjustment, high energy dissipation, low cost, and robustness to frequency variations. ...
Suspended piping systems often suffer from severe damages when subjected to seismic excitation. Due to the high flexibility of the piping systems, reducing their displacement is important for the prevention of damage during times of disaster. A solution to protecting piping systems during heavy excitation is the use of the emerging pounding tuned mass damper (PTMD) technology. In particular, the single-sided PTMD combines the advantages of the tuned mass damper (TMD) and the impact damper, including the benefits of a simple design and rapid, efficient energy dissipation. In this paper, two single-sided PTMDs (spring steel-type PTMD and simple pendulum-type PTMD) were designed and fabricated. The dampers were tested and compared with the traditional TMD for mitigating free vibration and forced vibration. In the free vibration experiment, both PTMDs suppressed vibrations much faster than the TMD. For the forced vibration test, the frequency response of the piping system was obtained for three conditions: without control, with TMD control, and with PTMD control. These novel results demonstrate that the single-sided PTMD is a cost-effective method for efficiently and passively mitigating the vibration of suspended piping systems. Thus, the single-sided PTMD will be an important tool for increasing the resilience of structures as well as for improving the safety of their occupants.
... Compared with other ways of damper, particle damping have many advantages such as simple structure, low costs, small modification of original structure, low additional mass and good adaptability to a wide temperature range [26]. Particle damping has been widely applied in broad engineering areas such as aerospace, automobile and precision machinery, etc. [22,27,28]. We aim to apply particle damping in SPAs vibration suppression. ...
... Impact, pounding, or collision between or among objects is an effective way to dissipate kinetic energy, including vibration energy. Impact dampers have been extensively studied for many years for vibration control (Li and Darby 2006;Bapa and Sankar 1985;Duncan et al. 2005;Li and Darby 2009;Semercigil et al. 1992;Ying and Semercigil 1991). It has been shown that impact dampers are relatively inexpensive and are robust and effective for a wide range of frequencies. ...
In the last few years, a new type of passive damper, the pounding tuned mass damper (PTMD), has been proposed and studied for passive structural vibration control. Experiments and numerical simulations verify the effectiveness of PTMDs. Studies have shown that PTMDs have a larger vibration dissipation capability and better robustness than traditional tuned mass dampers (TMDs). The design of a PTMD involves several parameters, and a numerical parametric study has been performed, but no experimental parametric study of a PTMD has been carried out. This paper experimentally studies the parameter sensitivity of the pounding tuned mass damper for a traffic signal structure. The control of free vibration and forced vibration at different harmonic frequencies are obtained to assess the sensitivity of the PTMD. This study considers four parameters: the viscoelastic (VE) material thickness, the gap between the delimiter and vibrating rod, the mass ratio, and the natural frequency of the PTMD. The parametric studies yield interesting findings that can guide future design of PTMDs.
... Dabei werden vorhandene Hohlräume z.B. in Maschinenständern mit Schüttgut (meist Sand) gefüllt [3]. Der Hauptdämpfungsmechanismus dieser "Schüttgut -Dämpfer" ist die Energiedissipation durch teilelastische Stöße der Partikel untereinander und mit der Behälterwand [4,5,6]. Partikelgefüllte metallische Hohlkugeln wurden entwickelt, um diesen Dämpfungsmechanismus für ZMW zu nutzen. ...
Partikelgefüllte metallische Hohlkugeln stellen die Grundbausteine dar, aus denen durch Kleben, Löten, Gießen und Sintern eine neue Klasse von Leichtbauwerkstoffen zur Körperschalldämpfung aufgebaut wird. Durch die verschiedenen Verbindungstechniken werden die Verbundwerkstoffe unterschiedlichen Anforderungen bezüglich Einsatzbedingungen, Herstellung und Kosten gerecht. Aufgrund ihrer geringen Masse und hohen Steifigkeit haben Leichtbaukonstruktionen eine nur sehr eingeschränkte Fähigkeit zur Dämpfung mechanischer Schwingungen. Deshalb wurde die Partikelfüllung als zusätzliches Dämpfungselement in die bekannten Leichtbauwerkstoffe aus metallischen Hohlkugeln integriert. Es wurde eine Methode zur Beurteilung einzelner Hohlkugeln entwickelt und mit der Simulation der Vorgänge im Kugelinneren begonnen. Das Masseverhältnis zwischen Partikelfüllung und Kugelschale hat einen starken Einfluss sowohl auf die Energiedissipation beim Aufprall einer Einzelkugel auf eine Unterlage als auch auf die Dämpfung der Eigenfrequenzen der Probekörper. Der Einfluss des Kugeldurchmessers hingegen ist klein; und für die Dicke der Kugelschale konnte kein Einfluss nachgewiesen werden. Im Vergleich zu Referenzmaterialien konnten Dämpfungswerte erreicht werden, die ca. 2 Größenordnungen über denen üblicherweise eingesetzter kompakter Metalle liegen. Nur Blei erreicht eine vergleichbare Dämpfung, allerdings ist dessen Dichte mehr als 5-mal so hoch wie die der Probekörper, die durch Eingießen der partikelgefüllten metallischen Hohlkugeln in eine Aluminiumlegierung hergestellt wurden.
... ✓ Passive damping methods are seismic isolation methods, viscous dampers, viscoelastic dampers, metallic dampers, tuned mass dampers, tuned liquid dampers and friction dampers [2]. ✓ Particle dampers (Figure 2) are used to tune vibration in the harsh environmental and working requirements, such as pressure, temperature changes, working period [3]. ...
Particle dampers are passive devices that can operate in harsh environments. The main mechanism is based on exciting multiple particles placed in an enclosure which is connected to the main structure subjected to the dynamic load. The popularity of this treatment increases with its compatibility with many different retrofit applications. Damping behaviour of the dampers is understood to be based on loss mechanisms which are internal and external frictions, energy transfers through the collisions of the particles with themselves and with the cavity wall of the enclosure. The energy dissipation is characterised by the parameters such as the geometry of dampers, filling ratio, volume of the cavity, material of the damper, excitation amplitude level. Nonlinearity of the particle dampers is a major area of interest. The purpose of this (aim of the) research is to study damping effect and energy dissipation in presence of multiple particle dampers. This includes understanding of the coupling between dampers and the periodicity effect in a periodically arranged array of dampers. One of the objectives of the study is to explore the dissipative effects by using numerical studies on COMSOL Multiphysics. Another objective is to conduct experiments to show the nonlinearity by changing the parameters and to explore the coupling effect of dampers on a vibrating beam.
... In addition, the optimal damping is affected by the mass ratio and the clearance. The damping characteristics of a vertical impact damper under a wide range of frequencies and multiple amplitudes was studied by Duncan et al. [101] via numerical simulation. ...
Particle damping, an effective passive vibration control technology, is developing dramatically at the present stage, especially in the aerospace and machinery fields. The aim of this paper is to provide an overview of particle damping technology, beginning with its basic concept, developmental history, and research status all over the world. Furthermore, various interpretations of the underlying damping mechanism are introduced and discussed in detail. The theoretical analysis and numerical simulation, together with their pros and cons are systematically expounded, in which a discrete element method of simulating a multi-degree-of-freedom structure with a particle damper system is illustrated. Moreover, on the basis of previous studies, a simplified method to analyze the complicated nonlinear particle damping is proposed, in which all particles are modeled as a single mass, thereby simplifying its use by practicing engineers. In order to broaden the applicability of particle dampers, it is necessary to implement the coupled algorithm of finite element method and discrete element method. In addition, the characteristics of experimental studies on particle damping are also summarized. Finally, the application of particle damping technology in the aerospace field, machinery field, lifeline engineering, and civil engineering is reviewed at length. As a new trend in structural vibration control, the application of particle damping in civil engineering is just at the beginning. The advantages and potential applications are demonstrated, whereas the difficulties and deficiencies in the present studies are also discussed. The paper concludes by suggesting future developments involving semi-active approaches that can enhance the effectiveness of particle dampers when used in conjunction with structures subjected to nonstationary excitation, such as earthquakes and similar nonstationary random excitations.
... [18] An impact damper's control performance is achieved by collisions between the freely moving mass and the container walls. [19] Therefore, a novel type of passive damper, pounding tuned mass damper (PTMD), was proposed by integrating the mechanisms of a TMD and an impact damper. [20] Previous studies have verified the PTMD's damping performance for an offshore jumper in air, [21] power transmission towers, [22] and a traffic signal pole. ...
Pounding tuned mass damper (PTMD) is a novel type of passive damper. The PTMD utilizes collisions or impacts of a tuned mass with viscoelastic materials to efficiently dissipate the vibration energy of primary structures. The previous studies have verified its effective damping performance on a full-scale subsea jumper and other structures in air. This paper presents the first-ever experimental verification of a submerged PTMD system for vibration control of pipelike structures underwater. To facilitate the experimental studies, a vertical vibration system consisting of 4 springs and a cylindrical steel pipe was designed and set up in a water tank. Furthermore, a numerical method considering the effect of the added mass is described to estimate the natural frequencies of a submerged cylindrical pipe. Therefore, experimental results demonstrate that the PTMD system is effective and efficient to suppress the forced vibrations of the submerged cylindrical pipe at the tuned frequency and is also robust over a range of detuning frequencies.
This chapter focuses on exploiting contact nonlinearities. Dry friction and unilateral contact are ubiquitous in science and technology. Their effect cannot be properly described in a linearized way. Dry friction is often the main cause for dissipation in structural dynamics. Impulsive unilateral interactions (impacts) are ideal for cross-scale energy transfer. Contact is easy to practically realize, but the dynamics of mechanical systems undergoing contact interactions is challenging to accurately predict. This chapter is divided into friction damping, a self-adaptive system and impact absorbers. Besides illustrating the use of strong nonlinearity in those engineering applications, this chapter presents useful methodology, including predictive modeling and numerical computation techniques, and the concept of Nonlinear Modes.
Tuned mass Dampers (TMDs) are extensively used passive control devices for high-rise buildings to mitigate undesirable responses. In practice, limit devices are frequently applied due to the large displacement of TMD, resulting in the emergence of piecewise nonlinearity. In this study, a TMD with piecewise stiffness (PSTMD) is developed to investigate the piecewise nonlinearity, including piecewise linear stiffness (PLS) and piecewise nonlinear stiffness (PNS). Initially, a novel equivalent continuous method (ECM) is proposed based on the principles of the least squares method (LSM) to approximate the PSTMD using a nonlinear TMD with Duffing stiffness. The equivalent nonlinear TMD is proven to be accurate enough to approximate the PSTMD. Subsequently, frequency response analysis is conducted on the PSTMD designed by linear method, results indicate that the linear method fails to design PSTMD. Thirdly, the optimal parameters of PSTMD are determined based on the ECM and the improved design method for nonlinear TMD proposed by the author. Results indicate that this novel design method effectively enhances the performance of PSTMD and mitigates the effect of piecewise nonlinearity. Finally, the influence of parameters of PSTMD on the optimal tuning frequency is investigated. The influence of damping ratio of PSTMD is explored by numerical simulation, with results indicating that employing the damping ratio suggested by traditional linear design method as the optimal parameter is reasonable. In summary, the novel method proposed in this study holds significance for the engineering design of TMDs with piecewise characteristics.
Boring is nothing but internal turning operation. As like external turning, the boring operation is
also widely required in manufacturing. The problem persists with boring operation when axial
distance of the internal dimension increases. To perform large hole machining operations the
overhang length of the boring bar also increases. As the overhang length of the boring bar increases
(especially for smaller cross-section boring bar) the amount of deflection of the boring bar also
increases, which results into unwanted vibrations. These unwanted vibrations will create chatter
marks on the machined surfaces and which affect the surface finish i.e. the surface finish obtained
is very poor. The present day industrial demand does not allow to supply the machined component
with improper surface finish. The Indian small-scale industries, which are the major vendors of
the major units will not be able to solve these problems with costlier solutions. In this research
work an attempt is made to find out the solution for the vibration problem of the boring operation
by using passive vibration damper technique. By mounting this passive damper on the standard
boring bar available in the market and using the standard practice of boring operation i.e. applying
routine machining parameters an experiment was conducted on reliable CNC turning center
machine. The results obtained were found satisfactory and indicates that passive damper can be
used for minimizing the effects of vibrations which ultimately results into enhancement of surface
finish.
In this paper, a new test-rig is designed to study the vibro-acoustic behavior of a simplified automotive transmission system. The aim is to experimentally reproduce the gear whine phenomenon in the gearbox, which is caused due to the transmission error without the need of rotating elements. Instead, an innovative method of excitation is proposed using a piezoelectric actuator. The test rig is
designed to be an economical solution to otherwise complicated setups. Experimental investigations of the gearbox housing accelerations and noise emission are done, and the setup successfully captures the accelerations with its associated harmonics, with values that are very close to the state of a real
rotating gearbox. In addition, a correlation is found between the housing accelerations and the emitted noise in certain frequency ranges. This setup will be used in future to test and integrate an active vibration control solution, to mitigate the generated vibrations and noise.
In the trend of light weighting of automotive gearboxes, structure-borne and air-borne noise are more significant, and this calls for smarter solutions to tackle this problem. Most vibrations from the gearbox arise from the gear meshing, which could go up to several kilohertz in frequency and can be detrimental to comfort of passengers. Active solutions to control vibrations related to the gear meshing under operation is the central theme of this research. As a first step in the experimental investigations, the behavior of the gearbox in modal and experimental conditions is studied to evaluate the parameters affecting the vibration response of the housing. The objective is to monitor the influence of these parameters on the system’s response in the frequency range of 1000-5000 Hz, which is the frequency range of interest for this study. Effect of transmitted load on the gearbox housing vibration level is studied using a non-rotational set-up to achieve comparable results with a rotating test-bench. (page 210 of full text file)
This paper investigates a vibratory system that employs Hertzian contact theory in terms of free vibration. Nonlinear impact dampers are created by using a spring, mass, and viscous damper model. The model is simulated in MATLAB software using the DAS method as a proper simulation framework. The Taguchi method is used to examine the efficacy of impact dampers for controlling structural vibrations of the structure. It is demonstrated that the optimal factors obtained can effectively suppress the structure’s unwanted vibrations. The analysis of variance is also used to indicate the most influential factor of dampers to demonstrate their ability to quench vibrations.
The bouncing motion of a spherical ball following its repeated inelastic impacts with a horizontal flat surface is analyzed. The effect of air resistance on the motion of the ball is accounted for by using the quadratic drag model. The effects of inelastic impacts are accounted for by using the coefficient of restitution, which is assumed to remain constant during repeated impacts. Also presented is an extension of the analysis allowing for a velocity-dependent coefficient of restitution. Closed-form expressions are derived for the velocity, position, maximum height, duration, and dissipated energy during each cycle of motion. The decrease of successive rebound heights in the presence of air resistance is more rapid for higher values of the launch velocity, because the drag force is stronger and acts longer. Air resistance can significantly affect the value of the coefficient of restitution determined in a dropping ball test. For a given number of rebounds, the energy dissipated by inelastic impacts is greater than the energy dissipated by air resistance, if the launch velocity is sufficiently small. The opposite is true for greater values of the launch velocity. The derived formulas are applied to analyze the bouncing motion of a ping pong ball, tennis ball, handball, and a basketball.
This paper investigates the effect of displacement constraints on the attenuation performance of tuned mass dampers (TMDs) used in boring and turning applications. A simplified piecewise-smooth mechanical model is investigated through time domain simulations and hybrid periodic orbit continuation, first under harmonic excitation, then under regenerative cutting load. A quasi-frequency response function is derived for impacting TMDs through composition of different families of period-1 orbits, then an acceptability map for turning is formulated based on the appearance of cutting-edge contact-loss and fly-over events. The bi-stable domain boundaries are determined through two parameter continuation of contact-loss grazing events. It is shown that in both cases arising rigid body collisions can significantly hinder TMD damping performance and lead to resonance problems or machine tool chatter.
When a vibrating system is damped with more than one type of damping, it is necessary to determine which of these types of damping are more effective to control the resonant response. In such a case, identification of damping parameters from the responses of a vibrating system becomes an important factor. Therefore when the system is damped due to coulomb friction, viscous friction, it is necessary to develop theoretical and experimental methods for identification of these damping parameters from the responses of the vibrating system. As such, for identification of coulomb and viscous friction (and also with particle damping) parameters, it is proposed to develop methods irresponsible for the control of resonant response of vibrating systems. The paper contains experimental setup and results about these different types of mechanical vibrations.
Mechanical vibration frameworks with thick and Coulomb erosion are of significance in the utilizations of elements and control issues. Erosion dampers are utilized in gas turbine motors, rapid turbo siphons, enormous adaptable space structures underside of railroad bogie, vehicle suspension frameworks and so on These dampers are utilized to diminish thunderous anxieties by giving sliding contact between focuses encountering relative movement because of vibration, subsequently scattering full vibration energy. Ordinary inactive damping strategies incorporate gooey liquid damping, swirl current damping, visco-versatile damping, Coulomb rubbing damping, molecule sway damping and so on Molecule Vibration Damping (PVD) is a mix of effect damping and contact damping. In a PVD, metal or earthenware particles or powders of little size (0.05 to 5 mm in distance across) are set inside pits inside or connected to the vibrating structure. Metal particles of high thickness, for example, lead or Tungsten give high damping execution because of dispersal of motor energy. Molecule damping includes the potential energy ingestions and dissemination through force trade between moving particles and vibrating dividers, grinding sway compensation. The issue of ID of substance of each kind damping in a given framework is a significant exploration zone. At the point when a vibrating framework is damped with more than one sort of models of damping, it is important to figure out which of these kinds of damping are more viable to control the resounding reaction. In such case, it is essential to distinguish damping boundaries from the reactions of a vibrating framework. Subsequently, when a framework is damped because of Coulomb contact, thick erosion, it is important to create trial strategies for distinguishing proof of these damping boundaries from the reactions of the vibrating system. As such, it is proposed to create techniques for ID of Coulomb and gooey grating (and furthermore with molecule damping) boundaries answerable for the control of thunderous reaction of vibrating frameworks.
Parçacık darbesi ile sönümleme, ana yapı üzerindeki boşluk veya boşluklara çok sayıda küçük boyutlu parçacıklar yerleştirilerek sistemdeki titreşim enerjisinin azaltıldığı pasif bir sönümleme yöntemidir. Titreşimi azaltılacak sistem üzerinde oluşturulan hücreler içine yerleştirilen çok sayıdaki küçük taneli parçacıklar titreşim esnasında hareket ederler ve hücre içerisinde birbirlerine ve hücre duvarlarına çarparlar. Bu sayede ana sistemin titreşimini bir miktar sönümlerler. Bu çalışmada, yatay doğrultuda zeminden tahrikli tek serbestlik dereceli bir yapının titreşim seviyesinin düşürülmesinde parçacık darbesi ile sönümleyicinin performansı incelenmiştir. Ana yapı üzerine açılan hücreler içerisine çok sayıda küresel parçacıklar yerleştirilmiştir. Parçacıkların birbiri ve hücre duvarları ile olan dinamik etkileşimini modellemek için Ayrık Elemanlar Yöntemi kullanılmış ve parçacık sayısına bağlı olarak çok sayıda doğrusal olmayan denklem takımı elde edilmiştir. Bu denklemlerin sayısal olarak çözülmesiyle sistemin zamana bağlı olarak titreşim genlikleri ve her parçacığın hücre içerisindeki hareketi elde edilmiştir. Yapılan sayısal uygulamalarda parçacıkların ana sistemin titreşimlerini önemli ölçüde sönümlediği görülmüştür.
The concept of particle damping could be traced back to 1937, when Paget [1] was studying the vibration attenuation problem of the turbine blades, during which he invented the impact damper.
Continually using the machine with the vibration may lead to a blood circulation failure to a hand. Vertical vibration and the torsional vibration are generated on the handle part of those machines. If the particle damper is attached on such a machine, an impact to act on a hand may be reduced. The particle damper is a device decreasing vibration by many particles which are filled into the container of the collision damper. It is difficult to experimentally observe the complex behavior of particles in the container since compressive forces, frictional forces, and impacts are generated between particles and the wall. Therefore, calculation of particle behavior by modeling a discrete body that contains the particles and the container is necessary. Discrete element method (DEM) might be one of the effective methods for its simulation. In this study, the torsional vibration is focused on and the characteristic of the damper examined through experiments and DEM simulation. In the experiment and simulation, many spherical particles as impactors were filled into a container. The container was given periodically a twist motion. The calculation results were compared with the experimental results by means of provided resonance curve. As a result, it turned out that the torsional vibration decreases when the selected particles which have a suitable particle diameter and a filling ratio were filled into the container.
According to the directions of European Union for nearly all machines noise control arrangements are needed (EU environmental noise directive 2002/49/EC). Additional e.g. automotive industry requires lightweight materials to reduce the total weight of their cars which led to significant reduction of the CO 2 emission. The combination of both requirements can't be fulfilled by any to date existing material. Cellular materials in particular are suitable for applications in lightweight constructions, but even those materials until now exhibit too low damping of structure-born sound. The damping behaviour of particles witch are filled in cavities are well known. Hence, a technology was developed to synthesize particle filled metal hollow spheres (pfMHS). Lightweight materials made with these particle-filled metal hollow spheres reach e.g. damping of structure-born sound and Young's modulus like lead by less then one-fifth of its density. Thanks to the multitude of parameters density, Young's modulus and damping are tuneable in a wide range. 1. Motivation Cellular materials are developed to fulfil the demands of Energy-and CO 2-reduction in applications by weight reduction and to reduce the material requirements in production. But cellular materials present additional properties like energy absorption, thermal insulation and damping of airborne sound. The mean mechanisms of damping of airborne sound are the reflection of acoustic waves and the elongation of their routes. Conventional cellular materials show only low damping of structure-borne sound. Unfortunately damping is decreasing with increasing Young's Modulus of the materials. The reason is that vibration energy and so the dissipated energy is proportional to the Young's Modulus. That means an additional damping mechanism is needed. Today structure-born sound is damped by heavy materials or polymers. All metals damp by the thermoelastic effect. It means the temperature of an elastic dilated grain
Vibration dampers are the first line of defense against shock and impacts sustained by mechanical and structural systems. Consequently, for decades, new impact damping technologies have been developed and applied in several engineering fields to attenuate undesired vibrations. Linear particle chain (LPC) impact dampers are the latest category of impact dampers being developed for the mitigation of unwanted vibrations in many systems. However, the challenges associated with prototyping such devices made their application in practical systems very limited. This paper proposes five innovative designs for the LPC impact dampers satisfying a wide range of industry needs in terms of efficiency, cost, and sustainability. The proposed designs are fabricated and tested under the same conditions to assess their efficiency in attenuating the vibration of a simple structure. Each design showed consistent behavior, but some designs outperformed others depending on the geometry, physical characteristics, and type of structure. The detailed design, experimental study, and time response comparisons are presented here to provide an initial study towards the development of practical sustainable LPC vibration dampers for real engineering applications.
In this paper, the empirical models for predicting the modal damping ratio (\(\xi )\) of impact-damped flexible beams (IDFB) via gene expression programming (GEP) are proposed. The experimental data used in training and testing phases of the GEP are obtained from the literature. The training and testing sets of the empirical models for the GEP are chosen from the database. The empirical models are developed for predicting the \(\xi \) of IDFB as functions of gap between vibrating mechanical system and impact damper (c), mass of particle (m), modal amplitude at the location of the damper (\({\varPhi }_\mathrm{d} )\), frequency of excitation (f), and peak value of the imaginary part of the frequency response functions (\(F_\mathrm{I} )\). The results of empirical models are compared with the results of experimental study and equation given in the literature. The results of empirical models for the \(\xi \) are in good agreement with the experimental results according to the results of equation given in the literature. The results of empirical models also reveal that GEP technique exhibits better performance to predict the \(\xi \) of IDFB.
Momentum exchange impact dampers (MEIDs) were proposed to control the shock responses of mechanical structures. They were applied to reduce floor shock vibrations and control lunar/planetary exploration spacecraft landings. MEIDs are required to control an object’s velocity and displacement, especially for applications involving spacecraft landing. Previous studies verified numerous MEID performances through various types of simulations and experiments. However, previous studies discussing the optimal design methodology for MEIDs are limited. This study explicitly derived the optimal design parameters of MEIDs, which control the controlled object’s displacement and velocity to zero in one-dimensional motion. In addition, the study derived sub-optimal design parameters to control the controlled object’s velocity within a reasonable approximation to derive a practical design methodology for MEIDs. The derived sub-optimal design methodology could also be applied to MEIDs in two-dimensional motion. Furthermore, simulations conducted in the study verified the performances of MEIDs with optimal/sub-optimal design parameters.
The technique of a bean bag damper has been effectively applied in many engineering fields to control the vibroimpact of a structural system. In this study, the basic parameters responsible for the design of an effective bean bag: the size of beans, the mass ratio of the bean bag to the structure to which it is attached, the clearance distance and the position of the bag, are studied by both theoretical and experimental analyses. These will provide a better understanding of the performance of the bean bag for optimisation of damper design. It was found that reducing the size of beans would increase the exchange of momentum in the system due to the increase in the effective contact areas. Within the range of mass ratios studied, the damping performance of the damper was found to improve with higher mass ratios. There was an optimum clearance for any specific damper whereby the maximum attenuation could be achieved. The position of the bag with respect to nodes and antipodes of the primary structure determined the magnitude of attenuation attainable. Furthermore, the limitations of bean bags have been identified and a general criteria for the design of a bean bag damper has been formulated based on the study undertaken. It was shown that an appropriately configured bean bag damper was capable of reducing the amplitude of vibration by 80% to 90%.
This paper presents a study of dissipation mechanisms of the Non-Obstructive Particle Damping (NOPD), based on a discrete element method (DEM). The goal of the work is to provide a theoretical basis for engineering applications of NOPD technology. Computer simulation is carried out for the coupled movement model of a single degree of freedom system of NOPD. A quantitative analysis describes the energy transference and energy dissipation of the particles in a collision process of the particle-primary system and the particle-particle system. The analysis reveals that the effectiveness of frictional dissipation is on the same numerical scale as that of collision energy dissipation. The energy dissipation by friction is greater for smaller size of particles. Furthermore, the NOPD technique has a wide frequency band for vibration reduction. The theoretical findings of the work are found to be in a good agreement with the previous experiment results.
A new-type impact damper consists of a bed of granular materials moving in a container mounted on the primary vibrating system. This report deals with the damping performance of the impact damper mounted on a multibody vibrating system. First, theoretical analysis is conducted. The equations of motion are developed for an equivalent single degree - of - freedom system and attached damper mass by using the normal mode approach. Results of the analysis are applied to the special case of a three degree-of-freedom system, and the effects of damper positions are studied. Numerdcal simulation and experimental studies are also conducted. Their results confirmed the validity of the theoretical analysis. © 1992, The Japan Society of Mechanical Engineers. All rights reserved.
An advanced passive damping technology has been developed and shown to be effective in a broad range of application areas. The technology is based on the use of low-density granular materials for reducing structureborne vibrations. In many cases, reduced structural vibration leads to a reduction in airborne noise radiated from the structure, which is important in some applications. Granular materials like sand and lead shot have been used in the past for structural damping, but the weight penalty of these materials has prevented wide-spread used. Damping with low-density granular materials utilizes the effects of low sound speed in such materials without the penalty of heavy weight. Commercial application areas for this technology include structural damping in the automotive, aerospace, heavy equipment, and consumer products industries.
The dynamic behavior of three different designs of a cantilever boring bar are compared by means of forced vibration tests. These are a bar with diametrically opposed flats machined along its length, a bar fabricated from laminates of steel and a damping compound, and a bar fitted with an impact damper. The impact damper boring bar is found to be the most effective, and an improved design, giving increases in stable metal removal rates of more than 100 percent, is outlined and tested. A theoretical analysis is presented for predicting the effectiveness of the impact damper with a spring supported impacting mass. This analysis enables the optimum mass ratio and gap setting of the damper to be selected for specified characteristics of the vibrating systems to which the damper is fitted.
The exact solution for the symmetric two-impacts-per-cycle motion of the impact damper is derived analytically, and its asymptotically stable regions are determined. The stability analysis defines the zones where the modulus of all the eigenvalues of a certain matrix relating conditions after each of two consecutive impacts is less than unity.
Focused research in the area of Multi-Particle Impact Damping (MPID) has resulted in new methods of characterization and prediction. An analytical method has been developed, based on the particle dynamics method, that uses characterized particle damping data to predict damping in structural systems. A methodology to design particle damping for dynamic structures will be discussed. The complete design methodology has been validated in proof-of-methodology testing on a structural component in the laboratory.
This paper is c oncerned with the stability of the various modes of vibration of the impact damper system described in Part 1†. Charts are presented showing the ranges of dynamically stable and kinematically viable periodic two-impact-per-cycle motion. The condition of kinematic viability ensures that the trajectory of the free mass remains within the container throughout the motion.
It is shown that in general stable, viable, two-impact-per-cycle motion of the system exists under resonant conditions of the main mass and that away from resonance, where any damper is ineffective, this type of motion is generally unstable.
This paper describes the theoretical and experimental work leading to the application of an impact damper to the image carrying cylinder of a web-fed printing press. The need for such a device arises from the desire to overcome a traditional limit on the operating speed of a press of this kind. This limit is the onset of an unacceptable anomaly in the finished print known as streaking. Streaking is caused by vibratory bending of the image carrying cylinders in a direction perpendicular to their axis of rotation. The selection of design parameters for an impact damper intended to reduce these vibrations was the objective of the study reported herein.
The paper presents measuring methods and measurement results of vibration damping of shot-filled containers with some empty volume as the shot damper clearance. The energy dissipation by shot depends on two phenomena: the internal and the external (container walls) impacts of shot particles associated usually with their internal and external friction. This is the working principle of the single-/multi-mass impact dampers and the shot vibration damper as well. It is shown in the paper that the additional loss factor introduced by the shot moving in the properly shaped container is on the order of half of shot mass reduced to the primary system modal mass. It may be even twice more if the energy dissipation by shot friction will be fully utilized. This depends mainly on the motion condition of shot inside a container. It was shown in the paper also that loose shot or better shot tightly packed in a plastic bag or mesh can fulfill this condition. Shot dampers can work efficiently in a broad range of acceleration and frequency and need not be maintained. Due to this it may be very convenient to apply them to structural vibration problems.
A fundamental study on impact dampers is reported in which the performance of impact dampers was investigated from free damped vibration generated when a step function input was supplied to a leaf spring with a free mass. The investigation showed that the damping capability of impact dampers results from collision between the free mass and the main mass. The frequency of the system with an impact damper varies with the natural frequency of the main vibratory system and the mass ratio. The optimum damping effect is achieved in combinations of the mass ratio and a clearance, i.e. motion extent of the free mass. The use of a free mass of only 25% of the main mass and a clearance of 0.6 mm, for example, can improve the damping capability of the main vibratory system at least 10 times or more, even though the clearance and the free mass are not adjusted to an amplitude of the main vibratory system. The critical amplitude where the impact damper does not function is determined by the natural frequency and the acceleration of gravity.
An experimental and analytical study is made of the performance of particle dampers under wide-band random excitation. A small model, provided with a nonlinear auxiliary mass damper, was used to investigate the major system parameters that influence the performance of particle dampers: total auxiliary mass ratio, particle size container dimension, and the intensity and direction of the excitation. It is shown that properly designed particle dampers, even with a relatively small mass ratio, can considerably reduce the response of lightly damped structures. An approximate analytical solution, which is based on the concept of an equivalent single unit-impact damper, is presented. It is shown that the approximate solution can provide an adequate estimate of the root-mean-square response of the randomly excited primary system when provided with a particle damper that is operating in the vicinity of its optimum range of parameters.
Presented herein is a novel passive vibration damping technique that is referred to as 'Non-Obstructive Particle Damping (NOPD).' The NOPD technique consists of making small diameter holes (or cavities) at appropriate locations inside vibrating structures and filling these holes to appropriate levels with particles which yield the maximum damping effectiveness for the desired mode (or modes). Powders, spherical shaped, metallic, non-metallic or liquid particles (or mixtures) with different densities, viscosities and adhesive or cohesive characteristics can be used.
The performance of an impact damper for controlling the vibration of a harmonically excited, hard Duffing's oscillator is investigated. Both elastic and inelastic collisions are considered. The optimum design is based around a low amplitude, stable solution predicted by the harmonic balance method. The final design is obtained and verified by numerical integration. Also, the use of an impact damper as an on-off migration controller for a guided transition from the resonance to the non-resonance branch of the solution is proposed. This scheme obviates the need of continuous, noisy operation of an impact damper.
Not Available Now at Department of Mechanical Engineering, McMaster University, Hamilton, Ontario, Canada.
The performance of a resilient bean bag and a conventional rigid impact damper are compared for sinusoidal external forces. The bean bag damper is not only a better attenuator of the resonant displacement of a lightly damped oscillator but the contact forces and the noise generated by collisions are also reduced. Two semi-empirical approaches are detailed to predict the dynamic behaviour of this new damper. They both produce quite accurate forecasts of the steady displacement of the oscillator. The deformation of the bean bag itself, however, is less predictable due to its sensitivity to the contact forces.
This paper, which examines the effect of gravity on the performance of an impact der, extends a previous paper (I)‡ concerned with the horizontal system. The analysis is in two parts: Part 1 which concerns itself with the calculation of all the theoretically possible steady-state motions with two impacts of the free mass, one at each end of its container, per cycle of the main mass; and Part 2§ which examines the dynamic stability and kinematic viability of the various steady-state solutions obtained in Part 1.
The impact damper consists of a small mass free to move within a rigid container attached to the main vibrating system. Vibration suppression is achieved through energy dissipation during impacts and through cancellation of the effect of the external exciting force by the cyclical reaction of the free mass on the main system.
The analysis shows that, in general, several different steady-state motions are theoretically possible for a given exciting frequency and that the effect of gravity on the motion of the system is to reduce the frequency range of two-impact-per-cycle motion. It is also shown that under the practically important condition of resonance the motion consists of two impacts of the free mass per cycle of the main mass. Outside the resonant condition, where the motion of the main mass is weak, the free mass may impact more or less than twice per cycle. This condition is not studied.
Particle impact damping (PID) is a means for achieving high structural damping by the use of a particle-filled enclosure attached to the structure in a region of high displacements. The particles absorb kinetic energy of the structure and convert it into heat through inelastic collisions between the particles and the enclosure, and amongst the particles. In this work, PID is measured for a cantilevered aluminium beam with the damping enclosure attached to its free end; lead particles are used in this study. The effect of acceleration amplitude and clearance inside the enclosure on PID is studied. PID is found to be highly non-linear. Perhaps the most useful observation is that for a very small weight penalty (about 6%), the maximum specific damping capacity (SDC) is about 50%, which is more than one order of magnitude higher than the intrinsic material damping of a majority of structural metals (O(1%)). Driven by the experimental observations, an elementary analytical model of PID is constructed. A satisfactory comparison between the theory and the experiment is observed. An encouraging result is that in spite of its simplicity, the model captures the essential physics of PID.
The asymptotically stable vibrations of a loaded oscillator colliding periodically with a rigid mass are described. Comparison of the numerical results with the few existing examples is encouraging but inconclusive. Better overall agreement is demonstrated with fairly comprehensive measurements from a specially built experimental rig. Impact motions are shown to be very sensitive to small fluctuations in the clearance between masses and the stiffness and loading of the oscillator near its linear, or collisionless, resonant frequency. The rigid mass is quite an effective damper at or just above this frequency condition.
An efficient impact damper consists of a bed of granular materials moving in a container mounted on a multibody vibrating system. This paper deals with the damping characteristics of a multidegree-of-freedom (MDOF) system that is provided with the impact damper when the damper may be applied to any point of the system. In the theoretical analysis, the particle bed is assumed to be a mass which moves unidirectionally in a container, and collides plastically with its end. Equations of motion are developed for an equivalent single-degree-of-freedom (SDOF) system and attached damper mass with use made of the normal mode approach. The modal mass is estimated such that it represents the equivalent mass on the point of maximum displacement in each of the vibrating modes. The mass ratio is modified with the modal vector to include the effect of impact interactions. Results of the analysis are applied to the special case of a three-degree-of-freedom (3DOF) system, and the effects of the damper parameters including mode shapes and damper locations are determined. A digital model is also formulated to simulate the damped motion of the physical system.
The dynamics of an inclinded impact pair is investigated. The base mass motion is harmonic, and the secondary mass is constrained to move in an inclined slot within the base mass. The dynamics of the secondary mass for alternating impacts is formulated in terms of a map over one period of the base motion. Steady state 2:1 motions, their stability and subsequent period doubling bifurcations are studied via this map. Results, presented in the form of stability plots as a function of the incline angle indicate that this simple system can exhibit complex behavior. Results are explained by using the local bifurcation theory of maps.
The influence of a specific discontinuous dynamic vibration absorber on the motion of a vibrating system is investigated.
Attention is paid to the effectiveness in the case of free vibrations and of vibrations due to sinusoidal excitations, where
structural damping is also taken into account. For certain configurations numerical results are given.
The single unit impact damper under free and forced vibrations is studied. The effects of mass ratio, coefficient of restitution, and gap size on the free vibrations are determined by simulating motion on the digital computer. Agreement of theoretical results with the present and previous experimental results in the free vibration state is good. In the study of forced motion, charts are developed, by using the closed form solution, showing optimum gaps and corresponding displacement amplitude reduction within the resonant frequency range. The optimum gap at resonance is not necessarily optimal at other frequencies.
A deceptively simple difference equation is derived which approximately describes the motion of a small ball bouncing vertically on a massive sinusoidally vibrating plate. In the case of perfect elastic impacts, the equation reduces to the “standard mapping” which has been extensively studied by physicists in connection with the motions of particles constrained in potential wells. It is shown that, for sufficiently large excitation velocities and a coefficient of restitution close to one, this deterministic dynamical system exhibits large families of irregular non-periodic solutions in addition to the expected harmonic and subharmonic motions. The physical significance of these and other chaotic motions which appear to occur frequently in non-linear oscillations is discussed.
A self-tuning impact damper is disclosed that absorbs and dissipates vibration energy in the blades of rotors in compressors and/or turbines thereby dramatically extending their service life and operational readiness. The self-tuning impact damper uses the rotor speed to tune the resonant frequency of a rattling mass to an engine order excitation frequency. The rating mass dissipates energy through collisions between the rattling mass and the walls of a cavity of the self-tuning impact damper, as well as though friction between the rattling mass and the base of the cavity. In one embodiment, the self-tuning impact damper has a ball-in-trough configuration with tire ball serving as the rattling mass.
Design procedure and charts for the impact damper
- P C Pinotti
- M M Sadek
P.C. Pinotti, M.M. Sadek, Design procedure and charts for the impact damper, Proceedings of the 11th
International Machine Tool and Die Research Conference, vol. A, 1970, pp. 181-195.
A self-tuning impact damper for rotating blades
- K P Duffy
- R L Bagley
- O Mehmed
K.P. Duffy, R.L. Bagley, O. Mehmed, A self-tuning impact damper for rotating blades, NASA Tech Briefs TSP
LEW-168333, 2001, pp. 1-15.
Effect of gravity on the performance of an impact damper
- Sadek