Abstract:
A nonlinear attachment induces energy to be transferred and/or to be pumped from a primary system or primary structure to the nonlinear attachment. The nonlinear attachment is an essentially nonlinear device and functions as a nonlinear energy sink. The nonlinear attachment attaches to the primary system or primary structure as a module and does not require connection to a ground. The energy is irreversibly pumped from the primary system or primary structure to the nonlinear attachment during transient resonance capture between the nonlinear attachment and the primary system or primary structure. The nonlinear attachment attenuates the energy of the primary system or primary structure. The nonlinear attachment locally confines the energy and/or dissipates the energy through a passive means and/or an active means.

Description:
BACKGROUND OF THE INVENTION 
   The present invention generally relates to a device, a system and a method for transferring energy. More specifically, the present invention relates to a device, a system and a method for pumping energy, such as, for example, vibrational energy from a main or a primary system and/or a main structure to an essentially nonlinear attachment, such as, for example, a nonlinear energy sink (hereinafter “NES”). The NES functions as an energy absorber, connects to the main structure as a module and requires no separate connection to a ground. Energy pumping is a one-way, irreversible transfer of the energy to the NES. As a result, the energy does not flow back to the main structure. Transferring the vibrational energy to the NES facilitates vibration and shock attenuation in the main structure following a disturbance, such as, for example, an externally induced disturbance. Further, after spatial confinements of the disturbance in the NES, vibrational energy is efficiently dissipated through a passive means and/or an active means. Moreover, the device resembles a narrow-band device, such as, for example, a classical vibration absorber or a tuned mass damper. However, the device and the system function as a broad-band absorber while attached only to the main structure. The broad-band absorber is derived from an essential nonlinearity of a connecting stiffness which may be achieved either with a mechanical spring or through an active control. 
   The device, the system and the method transfers energy and/or undesired motion from the main structure following the disturbance. The disturbance may be from, for example, a transient load, such as, for example, a shock or due to maneuvering of the main structure. Maneuvering the main structure typically results in a residual vibration, such as, for example, ringing. The disturbance may also be self-induced as in, for example, a fluid-structure interaction, resulting in a sustained large-amplitude motion, such as, for example, a limit cycle oscillation (hereinafter “LCO”). Generally, the LCO interferes with the performance of a primary role of the main structure. Three strategies for reducing the effect of the disturbance on the main structure are as follows: isolation which reduces the energy reaching the main structure from the disturbance; damping which dissipates the energy from the disturbance within the main structure; and absorption which removes the energy reaching the main structure from the disturbance via an auxiliary device. 
   Vibration isolation requires the main structure to be at least a single-degree-of-freedom system. An objective of vibration isolation is to reduce a natural frequency or frequencies of the main structure well below the lowest frequency of excitation. As a result, responses to disturbances are attenuated well above the highest natural frequency of interest. Vibration damping limits the magnitude of a resonant response in the steady-state and controls the peak response and decay rate in a transient state. 
   Vibration absorption requires a minimum of two degrees of freedom with one or more degrees of freedom constituting the main structure while another remaining degree of freedom is a vibration absorber. A passive vibration absorber is commonly known as a tuned mass damper, a passive mass damper, a tuned mass absorber and/or a passive mass absorber. 
   A harmonic disturbance is associated with a single frequency. The addition of a degree of freedom, such as, for example, the vibration absorber, may reduce and/or may attenuate a response of the main structure at or near the exciting frequency. Attaching a linear vibration absorber to the main structure and tuning the absorber such that its natural frequency is equal to the frequency of the excitation accomplishes the reduction and/or attenuation of the response of the main structure. A mass ratio between a mass of the absorber and a mass of the main structure is typically as small as possible. 
   When the frequency of the excitation is equal or nearly equal to the natural frequency of the linear vibration absorber, the response of the primary structure is small and the response of the linear vibration absorber is large. As a result, the response is localized to the linear vibration absorber at the driving frequency. However, near the two new natural frequencies of the main structure with the attached linear vibration absorber, the responses of the main structure and the linear vibration absorber are large. 
   An attenuation band between two resonant peaks is controlled by the mass ratio and damping coefficients of the main structure and the linear vibration absorber. Increased damping makes the main structure more robust to parametric variations and decreases the attenuation efficiency while a higher mass ratio broadens the attenuation band. The linear vibration absorber is primarily a steady-state device. The linear vibration absorber takes the energy input to the main structure at a single frequency and channels the energy to the linear vibration absorber. As a result, the linear vibration absorber protects the main structure. Small changes in the excitation frequency renders the device counter-productive if a new driving frequency is close to one of the two natural frequencies of the main structure and linear vibration absorber that bound the driving frequency. 
   For impulsive, wide-band loading, the linear vibration absorber has limited utility because the linear vibration absorber results in two resonant regions over which both the primary structure and the linear vibration absorber magnify the input. A nonlinear system can be exploited to improve performance of a vibration absorber beyond that of the linear system. 
   Nonlinear stiffness elements may improve attenuation characteristics of a vibration absorption system without increasing complexity and/or compromising economics. Nonlinear designs may be designed to give a spatial confinement or a localization and/or an energy pumping which enhances a capacity of the vibration absorption system to attenuate effects of unwanted broadband and/or narrowband disturbances. The energy pumping cannot be achieved by standard linear and/or nonlinear designs. 
   Other nonlinear vibration absorbers (hereinafter “NVAs”) have been developed, but none were based on the energy pumping concept. Further, the effectiveness of the NVAs in a shock and/or a vibration isolation of a primary structure has been poor except over narrow frequency ranges. The NVAs are designed to operate near linearized natural frequencies or under conditions of an internal resonance between the natural frequencies of the primary structure. A local design of the NVAs is different from nonlinear energy pumping on which the present invention is based. 
   Therefore, a need exists for a device, a system and a method which transfers the energy from the main structure to a device and/or an attachment during transient resonance captures. A single resonance capture begins with the main structure vibrating at a large amplitude while a motion of a NES mass is comparatively small. However, even at small displacements, a essentially nonlinear spring connecting the NES mass to the primary structure provides some coupling. As a result, the energy begins to flow to the NES while the amplitude of the NES motion increases. The stiffness of the nonlinear spring depends on the deflection of the nonlinear spring. An amplitude and a frequency of NES motion will exist at which the NES can resonate with the main structure. As a result, an impedance match is achieved between the primary structure and the NES, and the energy flows readily into the NES with an attendant reduction in the energy and/or vibration of the primary structure. As the energy is dissipated in the NES by, for example, a passive damper, the amplitude of the NES motion diminishes and the resonance capture and/or corresponding impedance match are lost. As a result, the flow of energy between the primary structure and the NES is greatly reduced. The energy in the NES is confined therein and/or is prevented from returning to the main structure. The NES dissipates the energy trapped therein. Depending upon the dynamics of the primary system, another resonance capture may be reached, and previous scenario repeats. 
   The transient resonance captures are distinctly different from internal resonance in coupled undamped systems. The internal resonance is a steady state phenomenon that occurs between coupled nonlinear oscillators with no damping and typically results in nonlinear beating whereby the vibrational energy is continuously exchanged between the coupled oscillators. Hence, no irreversible transfer of energy from one oscillator to another oscillator occurs. 
   Furthermore, a need exists for a device, a system and a method which is an essentially nonlinear module or attachment for attenuating vibrations in main structures and/or structures subjected to dynamic loads, such as, for example, wide-band or narrow-band loads. The device, the system and the method absorbs, confines and dissipates the energy from vibrations in the main structure. The device is an advantage over the present state of the art because the device offers a protective solution for large scale, complex, flexible structures subjected to broad-band excitation. 
   Additionally, a need exists for a device, a system and a method for pumping vibrational energy from a main structure to the device or the attachment, the NES. Further, a need exists for a device, a system and a method for transferring vibrational energy rapidly from a main structure to the device or the attachment, the NES. Still further, a need exists for a device, a system and a method for dissipating energy confined within the device or the attachment, the NES. Moreover, a need exists for a device, a system and a method for protecting a primary structure by pumping energy from a main structure following a disturbance to the device or the attachment, the NES. 
   SUMMARY OF THE INVENTION 
   The present invention generally relates to a device, a system and a method for transferring and dissipating energy and/or controlling vibrations. Further, the present invention relates to a device, a system and a method for transferring and dissipating energy, such as, for example, vibrational energy from a primary structure to an essentially nonlinear energy sink (NES). More specifically, the present invention relates to a device, a system and a method for attenuating vibrations in main structures or primary structures subjected to dynamic loads generated by external disturbances, such as, for example, shocks, earthquakes, aerodynamic forces, fluid-structure interactions and/or the like. The NES protects the primary structure by rapidly pumping the energy and/or vibrations from the primary structure to the NES and/or dissipating the energy therefrom. The NES, though locally applied, modifies the global behavior of the combined primary structure-NES system. 
   A glossary providing definitions for pertinent terminology is as follows: 
   Active damper is defined as a device that requires an exogenous energy source and a control system to perform a function as a damper. 
   Active essentially nonlinear spring is defined as a device that requires an exogenous energy source and a control system to perform a function as an essentially nonlinear spring. 
   Dissipation is defined as an energy loss to an environment, usually in a form of heat. 
   Essential nonlinearity is defined as a characteristic behavior between two variables, one independent and another dependent, such that the dependent variable undergoes no change over some range of the independent variables in a neighborhood where a value of the independent variable is zero. 
   Essentially nonlinear spring is a compliant element for which a force-displacement relation is nonlinearizable. The complaint element is a spring which, for small displacements, exerts no force. 
   Geometrically nonlinear is defined as a demonstration of a nonlinear force-displacement relation as a result of large configuration changes, such as, for example, the lines of action of internal forces during deformation. 
   Impedance is defined as a frequency-dependent property of a system or a structure that incorporates a mass, a damping and a stiffness and reflects a compliance of the system or the structure under a harmonic loading. 
   Irreversible transfer of vibrational energy is defined as a one-way movement of energy from a primary system or a primary structure to a module or an attachment. 
   Mass and/or inertia is defined as a property of matter that causes the matter to resist motion. 
   Modular is defined as a fully self-contained structure having external connection only to the vibrating primary structure. 
   Narrow-band is defined as occurring over a small range of frequencies. A sinusoid is an ideal narrow-band signal with energy only at the frequency of oscillation of the sinusoid. 
   Nearly essential nonlinearity is defined as a characteristic behavior between two variables, one independent and another dependent, such that the dependent variable undergoes negligible change over some range of the independent variables in a neighborhood where a value of the independent variable is zero. 
   Non-smooth force-displacement relationship is defined as a force-displacement relationship with one or more kinks, jumps or breaks within a specified range of application. 
   A primary system or a primary structure is defined as a system or a structure which seeks protection and may be simple or complex and linear or nonlinear. 
   Order of magnitude change is defined as a difference in integer parts of a base 10 logarithm of each of two numbers. 
   Passive damper is defined as a damping device, such as, for example, a viscous device, a frictional device and/or the like. The passive damper requires no exogenous energy source to perform a function of the damping device. 
   Passive essentially nonlinear spring is defined as an essentially nonlinear spring that requires no exogenous energy source to perform a function of the essentially nonlinear spring. 
   Protection of a structure is defined as a reduction of a vibratory response thereby increasing an operating life of the structure. 
   Rapid transfer of vibrational energy is defined heuristically as moving energy from a primary system or a primary structure to a module or an attachment over a time scale defined by a relatively small number of cycles of vibration of the primary system or the primary structure. 
   Resonance capture is defined as a dynamic phenomenon whereby a nonlinear attachment momentarily “locks” in an impedance match with a primary system or a primary structure to which the nonlinear attachment is attached. In essence, resonance capture is a situation when an instantaneous frequency of oscillation of the nonlinear attachment “locks” in resonance with a multiple or a fraction of a resonant frequency of the primary system or the primary structure. 
   Resonance capture cascade is defined as a series of resonance captures each occurring at a different frequency, usually at or near a resonant frequency of the primary system or structure. 
   Resonant frequency is defined as a frequency associated with a mass and a linearized stiffness of a system or a structure at which the system or the structure will vibrate if transiently excited. 
   Slope of the force-displacement relation is defined as a change in force per unit change in displacement. 
   Smooth force-displacement relation is defined as a force-displacement relationship without kinks, jumps or breaks within a specified range of application. 
   Vibration control is defined as employing some means to reduce, or at least bound, an amplitude of vibration of a system or a structure. Prevailing methods of the vibration control include vibration isolation, vibration damping, and/or vibration absorption. 
   Wide-band is defined as occurring over a broad range of frequencies. An impulse is an ideal wide-band signal with equal energy at all frequencies from zero to infinity. 
   To this end, in an embodiment of the present invention a vibration control device is provided. The device has a mass and means for dissipating energy wherein the means for dissipating energy is attached to the mass and the structure. Further, the device has a spring wherein the spring is essentially nonlinear wherein the spring is attached to the mass and the structure and further wherein the energy is irreversibly transferred from the structure to the device and is dissipated therein. 
   In an embodiment, the spring has a smooth force-displacement relation or a non-smooth force displacement relation. 
   In an embodiment, the device may be modular, an attachment to the structure or integral with the vibrating system or structure. 
   In an embodiment, the spring is nonlinearizable at a zero displacement condition due to the slope of a force-displacement characteristic being zero at that point. 
   In an embodiment, the spring is linearizable at a zero displacement condition due to the slope of the force-displacement characteristic being non-zero at a point. 
   In an embodiment, a stiffness of the spring at the zero displacement condition is orders of magnitude smaller than the stiffness of the spring at displacements comparable to a resonance capture between the vibration control device and the structure wherein linearized dynamics of the vibration control device are at a zero displacement condition or near the zero displacement condition and do not appreciably affect dynamics of the interaction between the vibration control device and the structure during the resonance capture. 
   In another embodiment of the present invention, device for attracting, absorbing and dissipating vibrational energy in a primary system or primary structure is provided. The device has an essentially nonlinear subsystem wherein the subsystem or substructure attaches to the primary system or primary structure or is integral to the primary system or the primary structure. 
   In an embodiment, the subsystem engages in a resonance capture at one or more resonant frequencies of the primary system or the primary structure. 
   In an embodiment, the subsystem transiently exhibits dynamic properties matching dynamic properties of the primary system or the primary structure. 
   In an embodiment, the subsystem freely draws the vibrational energy from the primary system or the primary structure therein when the subsystem transiently exhibits dynamic properties matching dynamic properties of the primary system or the primary structure. 
   In an embodiment, the subsystem is locally applied to the primary system or primary structure and modifies global dynamic properties of the primary system or the primary structure for the purpose of promoting vibrational energy imparted to the primary system or the primary structure to pass one-way and irreversibly to the device and reducing or eliminating the response of the primary system or primary structure. 
   In an embodiment, the device has an integral damping element attached between the subsystem and primary system or primary structure wherein the integral damping element dissipates sufficient vibrational energy to cause a mismatch in dynamic properties between the subsystem and the primary system or the primary structure and precludes remaining vibrational energy from freely flowing outward from the subsystem to the primary system or the primary structure. 
   In an embodiment, the subsystem sequentially seeks resonant frequencies of the primary system or the primary structure wherein the subsystem engages in a resonance capture with each resulting in resonance capture cascading. 
   In another embodiment of the present invention, an essentially nonlinear device for transferring energy from a structure is provided. The device has a mass and an essentially nonlinear spring extending from the mass wherein the spring is attached to the structure. Further, the device has a damping device connected to the mass and to the structure wherein the damping device has a variable and controllable dissipation rate wherein the damping device requires an external power source and dissipates the energy. 
   In an embodiment, the energy is dissipated by a combination of a passive means and an active means. 
   In another embodiment of the present invention, an essentially nonlinear device is provided. The device has a mass and an actuator connected to the mass. Further, the device has a control system connected to the actuator and an energy source capable of producing an essentially nonlinear behavior wherein the energy source is connected to the control system. 
   In an embodiment, the essential nonlinearity behavior is achieved by a combination of a passive means and an active means. 
   In another embodiment of the present invention, a method for transferring energy from a structure is provided. The method has the step of providing a device having an essentially nonlinear spring, a damper and a mass wherein the essentially nonlinear spring and the damper is connected to the mass. Further, the method has the steps of connecting the device to the structure with the essentially nonlinear spring and the damper and pumping the energy from the structure to the device via the essentially nonlinear spring and the damper wherein the pumping of the energy to the device is one-way and irreversible. 
   In an embodiment, the method has the step of absorbing the energy from the structure with the device. 
   In an embodiment, the method has the step of dissipating the energy from the structure with the device. 
   In an embodiment, the method has the step of integrally forming the structure with the device. 
   In an embodiment, the method has the step of making the damper integral to the device. 
   In an embodiment, the method has the step of eliminating the energy from the structure. 
   In an embodiment, the method has the step of engaging in resonance capture with a resonant frequency of the structure with the device. 
   In an embodiment, the method has the step of transiently matching impedances of the device and the structure. 
   In an embodiment, the method has the step of transiently changing the impedance of the device to preclude the energy from flowing from the device to the structure. 
   It is, therefore, an advantage of the present invention to provide a device, a system and a method for transferring energy which allows the energy to be pumped from a primary structure to the NES. 
   Another advantage of the present invention is to provide a device, a system and a method for transferring energy which provides the NES to pump energy from a primary structure to the NES. 
   And, another advantage of the present invention is to provide a device, a system and a method for transferring energy which protects a primary structure by pumping the energy from the primary structure to the NES. 
   Yet another advantage of the present invention is to provide a device, a system and a method for transferring energy which provides the NES to dissipate the energy and/or the vibrations confined therein. 
   A further advantage of the present invention is to provide a device, a system and a method for transferring energy which provides the NES to rapidly transfer the energy from a primary structure to the NES. 
   Moreover, an advantage of the present invention is to provide a device, a system and a method for transferring energy which provides the NES that does not require connection to a ground. 
   And, another advantage of the present invention is to provide a device, a system and a method for transferring energy which provides irreversible transfer of the energy from a primary structure to the NES. 
   Yet another advantage of the present invention is to provide a device, a system and a method for transferring energy which improves vibration and/or shock attenuation in the primary structure following externally induced disturbances. 
   Another advantage of the present invention is to provide a device, a system and a method for transferring energy which provides spatial localization of response due to disturbances of a primary structure to the NES. 
   Yet another advantage of the present invention is to provide a device, a system and a method for transferring energy over a broad-band of frequencies to the NES. 
   A still further advantage of the present invention is to provide a device, a system and a method for transferring energy which provides the NES which has an essential nonlinearity with respect to a primary structure. 
   Moreover, an advantage of the present invention is to provide a device, a system and a method for transferring energy which reduces response due to seismic shock in buildings, bridges and/or other civil engineering infrastructures. 
   And, another advantage of the present invention is to provide a device, a system and a method for transferring energy which provides a base isolation system for a building, a bridge and/or another civil engineering infrastructure. 
   Yet another advantage of the present invention is to provide a device, a system and a method for transferring energy which allows a transient disturbance to be dissipated via the NES. 
   A further advantage of the present invention is to provide a device, a system and a method for transferring energy which provides vibration suppression across a broadband of frequencies. 
   Moreover, an advantage of the present invention is to provide a device, a system and a method for transferring energy which allows nonlinear energy sinks to be applied to more than one floor of a building to dissipate response due to a seismic shock. 
   A further advantage of the present invention is to provide a device, a system and a method for transferring energy which modifies the global dynamics of the primary system or primary structure, which may result in stabilization of the system and reduction and/or elimination of, for example, limit cycle oscillations typical of self-excited systems, such as, for example, aircraft in flight, pipes with internal fluid flows and/or the like. 
   Additional features and advantages of the present invention are described in, and will be apparent from, the detailed description of the presently preferred embodiments and from the drawings. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a schematic diagram of a two-degree-of-freedom system with a single-degree-of-freedom primary structure and an essential nonlinear energy absorbing device in an embodiment of the present invention. 
       FIG. 2  is a graph of a percentage of total input energy dissipated at the essential nonlinear absorbing device of  FIG. 1  as a function of the total input energy in an embodiment of the present invention. 
       FIG. 3   a  is a graph of linear oscillator displacement-time for the transient responses of the primary structure of  FIG. 1  in an embodiment of the present invention. 
       FIG. 3   b  is a graph of linear oscillator displacement-time for the transient responses of the essential nonlinear energy absorbing device of  FIG. 1  in an embodiment of the present invention. 
       FIG. 4  is a schematic diagram of a multi-degree-of-freedom linear system with an essential nonlinear energy absorbing device attached thereto in an embodiment of the present invention. 
       FIG. 5   a  is a graph of relative transient response-time for the two-degree-of-freedom linear system of  FIG. 4  without the essential nonlinear attachment in an embodiment of the present invention. 
       FIG. 5   b  is a graph of instantaneous frequency-time for the essential nonlinear attachment of  FIG. 4  in an embodiment of the present invention. 
       FIG. 6  is a perspective view of a nonlinear energy sink attached to a primary structure in an embodiment of the present invention. 
       FIG. 7  is a perspective view of a nonlinear energy sink in an embodiment of the present invention. 
       FIG. 8  is a cross-sectional perspective view of a nonlinear energy sink in an embodiment of the present invention. 
       FIG. 9  is a schematic diagram of a nonlinear energy sink having multiple linear springs which are used to achieve a piecewise-linear, but overall nearly essentially or essentially nonlinear, coupling stiffness in an embodiment of the present invention. 
       FIG. 10  is a schematic diagram of a nonlinear energy sink having buckled columns which serve as nearly essentially or essentially nonlinear springs in an embodiment of the present invention. 
       FIG. 11  is a schematic diagram of a nonlinear energy sink having nearly essential or essential nonlinearity which is achieved by a variation in effective spring length that accompanies displacement of the sink mass in an embodiment of the present invention. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   The present invention generally relates to a device, a system and a method for transferring energy from a primary structure to a nonlinear energy sink (hereinafter “NES”). More specifically, the present invention relates to a device, a system and a method for energy absorption, dissipation and resulting vibration attenuation in primary systems and primary structures. The device, the system and the method are based on nonlinear energy pumping. Nonlinear energy pumping is a one-way, passive or active, irreversible transfer of vibrational energy initiated by an exogenous disturbance, such as, for example, a narrow-band force, a wide-band force and/or a system of forces, to a priori assigned modules or nonlinear attachments, such as, for example, the NES. The energy confined in the NES is thereby efficiently dissipated. A transient resonance capture governs and/or controls the nonlinear energy pumping. An instantaneous frequency of oscillation of the nonlinear attachment locks in resonance with an integer multiple of, or with a rational fraction of a resonant frequency of the primary system or the primary structure. As a result, energy is permitted to flow from the primary system or primary structure to the nonlinear attachment or the NES. 
   The present invention represents a new method for energy absorption and/or dissipation with a direct application to vibration attenuation and/or shock isolation in primary structures. The present invention differs from any other known devices in the field in concept and/or in the use of an essentially nonlinear stiffness element as a means for achieving the nonlinear energy pumping. The use of nonlinear energy pumping manifested through an entirely modular nonlinear energy sink to achieve the shock isolation and/or the vibration attenuation is completely new and cannot be considered as an improvement to any existing apparatus, device, product, system, process, method and/or composition of matter. 
   The present invention generally relates to a device, a system and a method for transferring energy. More specifically, the present invention relates to a device, a system and a method for transferring energy, such as, for example, vibrational energy from a system or structure to an attachment or module. The attachment or module is essentially nonlinear and may be the NES. Attachment of the NES to a primary system or structure allows the vibrational energy to be irreversibly transferred from the primary system or the primary structure to the NES. As a result, the NES promotes attenuation of vibrations in the primary system or the primary structure caused by external disturbances, such as, for example, wide-band disturbances and/or narrow-band disturbances. 
   The primary system or the primary structure exhibits resonant frequencies, often called resonances. The NES has no inherent resonant frequency due to an essential nonlinearity of the NES. Excitement of the primary system or the primary structure with sufficient energy causes the vibrational energy to flow to the NES which undergoes resonance capture with the primary system or the primary structure. The impedance of the NES then matches the impedance of the primary system or the primary structure. As a result, the vibrational energy flows to the NES until sufficient vibrational energy is dissipated by the NES causing an impedance mismatch which prevents the vibrational energy from flowing back to the primary system or the primary structure. This is energy pumping which is the one way, irreversible transfer of the vibrational energy from the primary system or the primary structure to an NES. As a result, the NES protects the primary system or the primary structure by pumping the vibrational energy from the primary system or the primary structure to the NES where it is eventually dissipated. 
   As the primary system or the primary structure exhibits more than one resonant frequency, the NES undergoes resonance capture at one resonant frequency, during which energy is pumped to the NES until the resonance capture is lost. The NES then inherently seeks another resonant frequency, engages in another resonance capture until insufficient vibrational energy remains in the primary system or the primary structure to promote additional resonance captures with the NES. This process of sequential, multiple resonance captures is referred to as resonance capture cascading. 
   An objective of the energy pumping by an addition of the NES to the primary system or the primary structure is a rapid, one-way, irreversible and nearly complete transfer of the vibrational energy from the primary system or the primary structure, subjected to the wide-band or narrow-band excitation, to the NES where the vibrational energy is dissipated. As a result, vibrations in the primary system or the primary structure are attenuated thereby protecting the primary system or the primary structure from damage by the vibrations. 
   Referring now to the drawings wherein like numerals refer to like parts,  FIG. 1  illustrates a schematic of a two-degree-of-freedom system  1 . The system  1  has a primary linear oscillator  10  and a nonlinear energy sink  20  (hereinafter “NES  20 ”). The primary linear oscillator  10  is a single-degree-of-freedom oscillator. The primary linear oscillator  10  is connected to a ground  11 . The primary linear oscillator  10  has a spring  13 , a viscous damper  12  and/or a quantity of matter, such as, for example, mass  14 . The mass  14  is, for example, a rigid mass. Further, the mass  14  may be, for example, a plate, a disk, a cube and/or the like. The mass  14  may be made from a material, such as, for example, steel, plastic, fiberglass, wood, cement and/or the like. The present invention should not be deemed as limited to the embodiments of a specific material of the mass  14 . 
   The NES  20  is, for example, an essentially nonlinear attachment. The NES  20  has a spring  22 , viscous damper  21  and/or a quantity of matter, such as, for example, a mass  23 . The mass  23  is, for example, a rigid mass. Further, the mass  23  may be, for example, a plate, a disk, a cube and/or the like. The mass  23  may be made from a material, such as, for example, steel, plastic, fiberglass, wood, cement and/or the like. The present invention should not be deemed as limited to the embodiments of a specific material of the mass  23 . 
   Here, the excitation given by F(t) may be impulsive and, thus, ideally a wide-band excitation. The mass  23  may be small with respect to the mass  14 . The absolute displacement of the mass  14  is given by y(t), and the absolute displacement of the mass  23  is given by v(t). The spring  13  connecting the mass  14  to the ground  11  is linear and the viscous damper  12  connects the mass  14  to the ground  11  in parallel with the spring  13 . The spring  22  is an ideal, smooth, essentially nonlinear spring with stiffness characteristic Cu 3 . The spring  22  connects the mass  14  to the mass  23  and u(t)=y(t)−v(t). The viscous damper  21  connects the mass  23  to the mass  14  in parallel with the essentially nonlinear spring  22 . The excitation F(t) is applied directly to the primary structure and/or the primary linear oscillator  10  with both the mass  14  and the mass  23  initially at rest. 
   An impulsive excitation of small magnitude applied to the mass  14  may result in an insignificant transfer of the vibrational energy to the NES  20 , comprised of the mass  23 , the essentially nonlinear spring  22 , and the viscous damper  21 . However, impulsive excitations of higher magnitude applied to the mass  14  may cause a significant transfer of the energy to the NES  20 , which engages in a 1:1 resonance capture with the primary linear oscillator  10 , given by the mass  14 , the viscous damper  12  and the spring  13 , at which time, an instantaneous resonant frequency of the NES  20  is approximately equal to the resonant frequency of the primary linear oscillator  10 . An impedance match occurs between the primary linear oscillator  10  and the NES  20 . As a result, the vibrational energy flows one-way and irreversibly to the NES  20  until there is adequate energy loss due to dissipation in the NES  20  to cause the NES  20  to escape the capture. Therefore, the vibrational energy is contained within the NES  20  and/or is eventually dissipated therein. 
   Certain periodic orbits are termed special orbits and are of particular importance in terms of irreversible energy transfer from the primary linear oscillator  10  to the NES  20 . The special orbits correspond to all initial conditions equal to zero with the exception of the initial velocity of the primary linear oscillator  10 . These initial conditions are identical to the initial conditions of the system  1  immediately after the primary linear oscillator  10  is excited by an impulsive force. 
   Specific special orbits are localized at the NES  20  and correspond to large-amplitude oscillations of the NES  20  and to simultaneous small-amplitude oscillations of the primary linear oscillator  20 . This implies that, if the system  1  is forced impulsively and one of the localized special orbits is excited, motion is rapidly and passively transferred from the primary linear oscillator  10  to the NES  20 . Therefore, the impulsive excitation of a special orbit may be a triggering mechanism for the initiation of the energy pumping. 
   The ability of the NES  20  to transfer the energy from the primary linear oscillator  10  to the NES  20  has been demonstrated through numerical simulation and physical experiments. The simulations were used to compute a portion of total input energy applied by an external impulse to the primary linear oscillator  10 , which is eventually dissipated by a dashpot of the NES  20 . A series of simulations were performed with parameters ε=0.05, ελ=0.01, ω 0   2 =1, C=1, F(t)=Yδ(t) and zero initial conditions. Impulsive excitation of magnitude Y was applied directly to the mass  14 . 
     FIG. 2  illustrates a graph of a percentage of total input energy dissipated at the NES  20  and is shown as a function of the total input energy. This measure quantifies the energy transfer from the primary linear oscillator  10  to the NES  20 . The total energy dissipated approaches a maximum percentage, such as, for example, ninety percent. 
     FIGS. 3   a  and  3   b  illustrate transient responses of the primary linear oscillator  10  and the NES  20 , respectively, which are depicted for parameters ε=0.05, ελ=0.0015, ω 0   2 =1, C=1, F(t)=−0.1039δ(t), with removal of the viscous damper  12  of the primary linear oscillator  10 . Energy pumping to the NES  20  is realized, as evidenced by the small amplitude of oscillation of the primary linear oscillator  10  as shown in  FIG. 3   a  with respect to the large amplitude of oscillation of the NES  20  as shown in  FIG. 3   b.    
   In the event that the primary structure possesses multiple resonant frequencies and associated modes of vibration, a broadband disturbance may excite more than one of these modes to undesirably large amplitudes. It is an advantage of the NES  20  that the NES  20  can interact with several modes of the primary structure in sequence and extract the vibrational energy from the primary structure. A sequence of extractions from several modes is called a resonance capture cascade and is demonstrated by using a system consisting of a 2-degree-of-freedom linear oscillator and an NES  20 . This is the simplest structural system capable of exhibiting a cascade response. 
     FIG. 4  illustrates a multi-degree-of-freedom system  100  with the NES  20 . The multi-degree-of-freedom system  100  has a 2-degree-of-freedom linear system  110  (hereinafter “2DOF system  110 ”) with the NES  120  in an embodiment of the present invention. For simulation purposes, the parameter values ω 0   2 =136.9, λ=0.155,  λ =0.544, d=1.2×10 3 , ε=1.8, and C=1.63×10 7  are assumed which correspond to the following two linear natural frequencies for the primary structure: ω 1 =11.68 rad/sec and ω 2 =50.14 rad/sec. The 2DOF system  110  is the primary structure, for example, a ground  111  and/or a ground  115 . The 2DOF system  110  may have viscous dampers  113 ,  117  and/or linear springs  112 ,  116  and/or masses  114 ,  118 , and is connected to ground  111  and/or ground  115 . The masses  114 ,  118  are, for example, rigid masses. Further, the masses  114 ,  118  may be, for example, plates, disks, cubes, a combination thereof and/or the like. The masses  114 ,  118  may be made from a material, such as, for example, steel, plastic, fiberglass, wood, cement and/or the like. The present invention should not be deemed as limited to the embodiments of a specific material of the masses  114 ,  118 . 
   The NES  120  is, for example, an essentially nonlinear attachment. The NES  120  has a spring  121 , viscous damper  122  and/or a quantity of matter, such as, for example, a mass  123 . The spring  121  is an essentially nonlinear spring. The mass  123  is, for example, a rigid mass. Further, the mass  123  may be, for example, a plate, a disk, a cube and/or the like. The mass  123  may be made from a material, such as, for example, steel, plastic, fiberglass, wood, cement and/or the like. The present invention should not be deemed as limited to the embodiments of a specific material of the mass  23 . 
   The 2DOF system  110  is attached to the ground  111  via the linear spring  112  and the viscous damper  113  and/or is attached to the ground  115  via the linear spring  116  and the viscous damper  117 . The mass  114 , the linear spring  112 , viscous damper  113  and/or the ground  111  is connected to the mass  118 , the linear spring  116 , the viscous damper  117  and/or the ground  115  via a spring  119 . The NES  120  is attached to the 2DOF system  110  via the essentially nonlinear spring  121  and the viscous damper  122  of the NES  120 . Furthermore, the NES  120  is connected to the masses  114 ,  118 , the linear springs  112 ,  116 , the viscous dampers  113 ,  117 , the ground  111  and/or the ground  115  via the essentially nonlinear spring  121  and the viscous damper  122  of the NES  120 . 
   Here, the excitation given by F(t)may be impulsive and, thus, ideally a wide-band excitation. The mass  123  may be small with respect to the masses  114 ,  118 . The absolute displacement of the masses  114 ,  118  is given by y 1 (t) and y 2 (t), respectively, and the absolute displacement of the mass  123  is given by v(t). The spring  119  and the essentially nonlinear spring  121  connects the masses  114 ,  118  to the mass  123  where u(t)=v(t)−y 2 (t). The excitation F(t) is applied directly to the 2DOF system  110  with the masses  112 ,  118  and the mass  123  initially at rest. 
   Impulsive excitations of small magnitude are applied to the masses  112 ,  118 , respectively may result in an insignificant transfer of the vibrational energy to the NES  120 . However, impulsive excitations of higher magnitude applied to the masses  112 ,  118  may cause a significant transfer of the energy to the NES  120 , which engages in a 1:1 resonance capture with one and/or both modes of vibration of the 2DOF system  110 , at which time, an instantaneous resonant frequency of the NES  120  is approximately equal to a resonant frequency of the 2DOF system  110 . An impedance match occurs between the 2DOF system  110  and the NES  120 . As a result, the vibrational energy flows one-way and irreversibly to the NES  120  until there is adequate energy loss due to dissipation in the NES  120  to cause the NES  120  to escape the capture. Therefore, the vibrational energy is contained within the NES  120  and/or is eventually dissipated therein. 
     FIG. 5   a  illustrates a graph of relative transient response (time history) for the system without the NES  120  in an embodiment of the present invention. Further,  FIG. 5   a  illustrates a relative response v(t)−y 0 (t) of the system  100  for initial displacements y 0 (0)=0.01, y 1 (0)=v(0)=−0.01 and zero initial velocities. 
     FIG. 5   b  illustrates a graph of instantaneous frequency content of a transient response of the NES  120 . Dashed line  202  represents the lower linear natural frequency and dashed line  204  represents the higher linear natural frequency. Ranges  211 ,  212 ,  213 ,  215 ,  216 ,  218  represent strong nonlinear interactions of the NES  120  with different low-frequency and high-frequency nonlinear modes of the system  100 . Range  214  represents a resonance capture with the linearized in-phase mode of primary linear oscillator  110 . Range  217  represents a resonance capture with the linearized out-of-phase mode of the primary linear oscillator  110 . As the energy decreases due to damping, a series of eight resonance captures is observed. 
   During each resonance capture the NES  20  passively absorbs energy from the nonlinear mode involved before escape from resonance capture occurs and the NES  120  transiently resonates with the next mode in the series. The NES  120  acts as a passive, broadband boundary controller, by, for example, absorbing, confining and/or eliminating the vibrational energy from the primary linear oscillator  110 . The capacity of the NES  120  to engage in the resonance captures with the multiple linear modes and the nonlinear modes in different frequency ranges is caused by the essential nonlinearity of the NES  120 ; the absence of a linear term in the stiffness characteristic precludes any preferential resonant frequency. 
   An example of this phenomenon occurs with primary structures with more than two degrees of freedom which include continuous structures with infinitely many degrees of freedom. A broadband disturbance typically imparts energy to multiple modes of the primary structure whereby a single NES  20  can extract the energy from multiple modes at differing resonant frequencies. The mechanics of an interaction of the NES  20  with each mode is essentially the same as the mechanics of an interaction of an NES  20  with a single-degree-of-freedom primary structure as described above. 
   The NES  20  has an ability to absorb energy from a sequence of vibratory modes at differing frequencies which is an important advantage of the NES  20 . It follows from an ability of the NES  20  to engage in transient resonant capture with the primary structure and/or the primary linear oscillator  10  at an arbitrary frequency. As a result, the ability of the NES  20  to absorb energy from a sequence of vibratory modes at differing frequencies depends upon the essentially and/or nearly essentially nonlinear force-displacement characteristic of the stiffness of the spring  22  of the NES  20 . 
     FIG. 6  illustrates a nonlinear energy sink  500  (hereinafter “NES  500 ”) which may be attached to a primary structure  502  or a the primary linear oscillator  10  in an embodiment of the present invention. The primary structure  502  has a first end  501  and/or a second end  503 . The first end  501  is opposite to the second end  503 . The primary structure  502  may be, for example, a piece of material, such as, for example, aluminum, steel, cement, wood and/or the like. The primary structure  502  may be, for example, on top of a horizontal air track (not shown in the drawings). Further, the primary structure  502  may support the NES  500  and/or may constrain the NES  500 . Moreover, the primary structure  502  may be, for example, the primary linear oscillator  10  and/or the like. The present invention should not be deemed as limited to the embodiments of a specific material of the primary structure  502 . 
   The NES  500  is connected to the ground  11  with, for example, a leaf spring  504  as shown in  FIG. 6 . The leaf spring  504  is, for example, a linear spring, such as, for example, the spring  13  of the primary linear oscillator  10  and/or the like. The leaf spring  504  has a grounding bolt  509  thereon. The grounding bolt  509  attaches to the ground  11 . As a result, the NES  500  is a single-degree-of-freedom (SDOF) structure which may oscillate along, for example, a line parallel to the primary structure  502 . A first upright  504  is attached to the first end  501  of the primary structure  502 . A second upright  506  is attached to the second end  503  of the primary structure  502 . 
   A first bearing block  505  is attached to the first upright  504 . A second bearing block  507  is attached to the second upright  506 . The first bearing block  505  extends inward with respect to the second upright  506 . The second bearing block  507  extends inward with respect to the first upright  504 . The first bearing block  505  and/or the second bearing block  507  is interposed between the first upright  504  and the second upright  506 . Each of the first bearing block  505  and/or the second bearing block  507  have a passage  508  therein. The passage  508  has, for example, a ball bearing therein. 
   As illustrated in  FIG. 6 , the first bearing block  505  and/or the second bearing block  507  may be, for example, substantially parallel to the primary structure  502 . The linear leaf spring  504  is attached to the second upright  506  of the primary structure  502 . As a result, the linear leaf spring  504  is connected to the primary structure  502  via the second upright  506 . 
   A shaft  510  extends from the first upright  504  to the second upright  506 . The shaft  510  passes through the passage  508  of the first bearing block  505  and/or the second bearing block  507 . As a result, the shaft  510  may be, for example, substantially parallel to the primary structure  502 . Further, the shaft  510  extends through the second upright  506  outward with respect to the primary structure  502 . 
   A mass  512  is connected to the shaft  510  via a mass retaining collar  514 . Still further, the mass  512  is interposed between the first upright  504  and the second upright  506 . The mass retaining collar  514  constrains the mass  512  to the shaft  510  and/or connects the mass  512  to the primary structure  502 . The mass  512  is, for example, the mass  23  or the NES  20 . The mass  512  may be, for example, a plate, a disk, a cube and/or the like. The mass  512  may be made from a material, such as, for example, steel, plastic, fiberglass, wood, cement and/or the like. The present invention should not be deemed as limited to the embodiments of a specific material of the mass  512 . 
   A support beam  516  is attached to the second upright  506  as shown in  FIG. 6 . The support beam  516  has a first end  518  and/or a second end  520 . The first end  518  is opposite to the second end  520 . The support beam  516  is interposed between the first upright  504  and the second upright  506 . The support beam  516  is adjacent to the leaf spring  502 . Further, the shaft  510  extends through the support beam to the second upright  506 . A first support block  522  and a second support block  524  are attached to the first end  518  and the second end  520 , respectively, of the support beam  516 . The first support block  522  and the second support block  524  extend outward with respect to the primary structure  502 . 
   A first clamp plate  530  and/or a second clamp plate  532  is connected to the shaft  510 . A first retaining collar  534  and/or a second retaining collar  536  is attached to the shaft  510 . The first clamp plate  530  and/or the second clamp plate  532  is interposed between the first retaining collar  534  and the second retaining collar  536 . 
   A first wire  526  and/or a second wire  528  extend from the first support block  522  to the second support block  524 . The first wire  526  and/or the second wire  528  extend between the first clamp plate  530  and the second clamp plate  532 . A first support cover  538  is attached to the first support block  522 . The first wire  526  and/or the second wire  528  is interposed between the first support cover  538  and the first support block  522 . As a result, the first wire  526  and/or the second wire  528  is attached to the first support block  522 . 
   A second support cover  540  is attached to the second support block  524 . The first wire  526  and/or the second wire  528  is interposed between the second support cover  540  and the second support block  524 . As a result, the first wire  526  and/or the second wire  528  is attached to the second support block  524 . The first wire  526  and/or the second wire  528  is connected to the support beam  516  and/or the primary structure  502  via the support blocks  522 ,  524  and/or the support covers  538 ,  540 . Further, the first wire  526  and/or the second wire  528  is connected to the shaft  510  and/or the mass  512  via the clamp plates  530 ,  532  and/or the retaining collars  534 ,  536 . 
   The NES  500  of  FIG. 6  includes the shaft  510 , the mass  512 , the first wire  526  and/or the second wire  528 . The NES  500  may be an essentially nonlinear spring, such as, for example, the NES  20 . The wires  526 ,  528  are deflected by motion of the shaft  510  with respect to the primary structure  502 . The wires  526 ,  528  provide, for example, a cubic nonlinear restoring force. Further, the support blocks  522 ,  524  may be moved along the support beam  516 . As a result, the stiffness of the essentially nonlinear spring may be increased or decreased. 
   The primary structure  502  may be excited by a broadband force. The broadband may be applied to the first upright  504  at the first end  501  opposite the leaf spring  502 . The broadband force induces energy in the primary structure  502 . As a result, the primary structure  502  may vibrate. The energy of the primary structure  502  is passively or actively absorbed by the NES  500  via the shaft  510 , the mass  512  and the wires  526 ,  528 . The energy of the primary structure  502  is transferred to the NES  500  or the mass  512 , the shaft  510  and the wires  526 ,  528 . As a result, the energy causes the mass  512  and/or the shaft  510  to move in a direction which may be substantially parallel to the primary structure  502 . Further, the energy is dissipated from the primary structure via the NES  500  or the shaft  510 , the mass  512  and/or the wires  526 ,  528 . This dissipation may be due to viscous losses in the shaft bearings or may occur primarily in a dashpot or other damper (not shown in the drawing) connected between the shaft  510  and the upright  504 , for example. 
     FIG. 7  illustrates a nonlinear energy sink  300  (hereinafter “NES  300 ”) in an embodiment of the present invention. The NES  300  has a frame  302 , a wire mesh  304  and/or a rigid mass  306 . The mass  306  may be, for example, a plate, a disk, a cube and/or the like. The rigid mass  306  may be made from a material, such as, for example, steel, plastic, fiberglass, wood, cement and/or the like. The present invention should not be deemed as limited to the embodiments of a specific material of the mass  306 . 
   The rigid mass  306  is attached to the wire mesh  304 . The rigid mass  306  is, for example, centered on the wire mesh  304  with respect to the frame  302 . As a result, the rigid mass  306  is connected to the frame  302  via the wire mesh  304 . The NES  300  may be, for example, the NES  20 . As a result, the wire mesh  304  is, for example, the spring  22 . Further, the rigid mass  306  may be, for example, the mass  23 . 
   The frame  302  has vertical walls  301   a ,  301   b  and/or horizontal walls  303   a,    303   b.  The wire mesh  304  has a plurality of vertical wires  305  and/or a plurality of horizontal wires  306 . The plurality of vertical wires  305  extend from the vertical wall  301   a  to the vertical wall  301   b.  The plurality of horizontal wires  307  extend from the horizontal wall  303   a  to the horizontal wall  303   b.  As a result, the frame  302  constrains the wire mesh  304 . Moreover, the plurality of vertical wires  305  may be, for example, substantially perpendicular with respect to the plurality of horizontal wires  307 . Moreover, geometrically nonlinear transverse deformation of the wire mesh  304  provides an essentially nonlinear or a nearly essentially nonlinear coupling between the mass  306  and the primary structure. 
   The frame  302  is attached to a primary structure (not shown in the drawings), such as, for example, the primary linear oscillator  10  and/or the like. The wire mesh  304  is undeformed when at rest and/or may be substantially parallel to the structure. The wire mesh  304  is stretched by the rigid mass  306  when the rigid mass  306  vibrates with respect to the frame  302 . Further, the rigid mass  306  has a direction of motion  320 . The direction of motion  320  may be, for example, along a normal with respect to the wire mesh  304  and/or the frame  302 . 
   As the broadband force is applied to the primary structure, the primary linear oscillator  10  and/or the frame  302 , the broadband force induces energy in the primary structure, the primary linear oscillator  10  and/or the frame  302 . As a result, the primary structure, the ground  11 , the primary linear oscillator  10  and/or the frame  302  vibrates. The energy of the primary structure, the primary linear oscillator  10  and/or the frame  302  is absorbed by the NES  300  via the wire mesh  304  and the rigid mass  306 . The energy of the primary structure, the primary linear oscillator  10  and/or the frame  302  is transferred to the NES  300  via the wire mesh  304  and/or the rigid mass  306 . As a result, the vibrational energy causes the rigid mass  306  to move in the direction of motion  320 . The vibrational energy is dissipated from the primary structure, the primary linear oscillator  10  and/or the frame  302  via the wire mesh  304  and/or the rigid mass  306 . Energy transferred to the NES  300  may be ultimately dissipated, for example, through the displacement of air or another surrounding fluid by the motion of the mass  306  and/or the wire mesh  304 , and/or through frictional losses (e.g., rubbing) at the wire interfaces. 
     FIG. 8  illustrates a nonlinear energy sink  400  (hereinafter “NES  400 ”) in an embodiment of the present invention. Further, the NES  400  has a cylinder  402 , a first plate  404 , a second plate  406  and/or a rigid mass  408 . The cylinder  402  has an interior surface  414  and/or an exterior surface  416 . The interior surface  414  is opposite to the exterior surface  416 . Still further, the NES  400  may be, for example, the NES  20 . As a result, the cylinder  402  may be the primary linear oscillator  10 . The rigid mass  408  may be, for example, a plate, a disk, a cube and/or the like. The rigid mass  408  may be made from a material, such as, for example, steel, plastic, fiberglass, wood, cement and/or the like. The present invention should not be deemed as limited to the embodiments of a specific material of the mass  408 . 
   The first plate  404  and/or the second plate  406  may have a shape, such as, for example, a circle. The first plate  404  and the second plate  406  have outer edges  410 ,  412 , respectively. The first plate  404  is connected to the second plate  406 . Further, the first plate  404  is buckled against the second plate  406 . Further, the first plate  404  and the second plate  406  may be the spring  22  of the NES  20 . 
   The rigid mass  408  is connected to the first plate  404 . The cylinder  402  is mounted to the primary structure, the primary linear oscillator  10  and/or the like. The outer edges  410 ,  412  of the first plate  404  and the second plate  406 , respectively, contact the interior surface  414  of the cylinder  402 . As a result, the cylinder  402  constrains the first plate  404  and the second plate  406  within the cylinder  402 . The rigid mass  408 , the first plate  404  and/or the second plate  406  have a direction of motion  418 . The direction of motion  418  may be substantially parallel with respect to the cylinder  402 . A nonlinear stiffness of the NES  400  is achieved by a radially preloaded, initial deformation of the first plate  404  and the second plate  406 . 
   As the broadband force is applied to the cylinder  402  and/or the flat plate, the broadband force induces energy in the cylinder  402  and/or the flat plate. As a result, the cylinder  402  and/or the flat plate vibrate. The energy of the cylinder  402  and/or the flat plate is absorbed by the NES  400  via the first plate  404 , the second plate  406  and/or the rigid mass  408 . The energy of the cylinder  402  and/or the flat plate is transferred to the first plate  404 , the second plate  406  and/or the rigid mass  408 . As a result, the energy causes the rigid mass  408  to move in the direction of motion  418 . The energy is dissipated from the cylinder  402  and/or the flat plate via the first plate  404 , the second plate  406  and/or the rigid mass  408 . Energy transferred to the NES  400  may be ultimately dissipated, for example, through the displacement of air or another surrounding fluid by the motion of the mass  408  and/or the plates  404 ,  406 , and/or through frictional losses (e.g., rubbing) at the interfaces of the plates, mass and/or cylinder. 
     FIG. 9  illustrates a nonlinear energy sink  600  (hereinafter “NES  600 ”) in an embodiment of the present invention. Further, the NES  600  has a housing  620 , a mounting flange  622 , guides  624 ,  625  and/or a nonlinear energy sink mass  626  (hereinafter “NES mass  626 ”). The guides  624 ,  625  connect the NES mass  626  to the housing  620  and constrain the NES mass  626  inside of the housing  620 . The NES  600  is attached to a primary structure  610  via the mounting flange  622 . The NES mass  626  may be, for example, a plate, a disk, a cube and/or the like. The NES mass  626  may be made from a material, such as, for example, steel, plastic, fiberglass, wood, cement and/or the like. The present invention should not be deemed as limited to the embodiments of a specific material of the mass  626 . 
   Weak linear centering springs  630 ,  631  are interposed between the NES mass  626  and the housing  620 . Strong linear springs  632 ,  633 ,  634 ,  635  are attached to the housing  620  and do not contact the NES mass  626  inside of the housing  620 . The strong linear springs  632 ,  633 ,  634 ,  635  have caps  642 ,  643 ,  644 ,  645 , respectively. The NES mass  626  will only contact the strong linear springs  632 ,  633 ,  634 ,  635  after the NES mass  626  is displaced across a gap from equilibrium. The weak linear centering springs  630 ,  631  are used to establish a nominal static equilibrium position of the NES mass  626  with respect to the housing  620 . If the weak linear centering springs  630 ,  631  are weak compared to the strong linear springs  632 ,  633 ,  634 ,  635 , a piecewise linear stiffness characteristic of the NES  600  is nearly essentially nonlinear. As a result the NES  600  is simple and easy to control the nonlinear stiffness through selection of the strong linear springs  632 ,  633 ,  634 ,  635  and adjustment of the gaps. 
   As the broadband force is applied to the housing  620  and/or the primary structure  610 , the broadband force induces energy in the housing  620  and/or the primary structure  610 . As a result, the housing  620  and/or the primary structure  610 . The energy of the housing  620  and/or the primary structure  610  is absorbed by the NES  600  via NES  600 , the weak linear centering springs  630 ,  631  and/or the strong linear springs  632 ,  633 ,  634 ,  635 . The energy of the primary structure  610  and/or the housing  620  is transferred to the weak linear centering springs  630 ,  631  and/or the strong linear springs  632 ,  633 ,  634 ,  635 . As a result, the energy causes the NES mass  626  to move in the direction of motion  601 . The energy is dissipated from the primary structure  610  and/or the housing  620  via the NES mass  626 , the weak linear centering springs  630 ,  631  and/or the strong linear springs  632 ,  633 ,  634 ,  635 . Energy transferred to the NES  600  may be ultimately dissipated, for example, through the displacement of air or another surrounding fluid by the motion of the NES mass  626 , the weak linear centering springs  630 ,  631  and/or the strong linear springs  632 ,  633 ,  634 ,  635 . 
     FIG. 10  illustrates a nonlinear energy sink  700  (hereinafter “NES  700 ”) in an embodiment of the present invention. Further, the NES  600  has a housing  720 , a mounting flange  722 , guides  724 ,  725  and/or a nonlinear energy sink mass  726  (hereinafter “NES mass  726 ”). The guides  724 ,  725  connect the NES mass  726  to the housing  720  and constrain the NES mass  726  inside of the housing  720 . The NES  700  is attached to a primary structure  710  via the mounting flange  722 . The NES mass  726  may be, for example, a plate, a disk, a cube and/or the like. The NES mass  726  may be made from a material, such as, for example, steel, plastic, fiberglass, wood, cement and/or the like. The present invention should not be deemed as limited to the embodiments of a specific material of the mass  726 . 
   Buckled columns  730 ,  731 ,  732 ,  733  are interposed between the NES mass  726  and the housing  720 . The buckled columns  730 ,  731 ,  732 ,  733  act as, for example, nonlinear springs. Stoppers  740 ,  742  are inside the housing  720  and enhance a hardening of the buckled columns  730 ,  731 ,  732 ,  733 . A stiffness of each of the buckled columns  730 ,  731 ,  732 ,  733  is small until the lateral motion of each of the buckled columns  730 ,  731 ,  732 ,  733  contacts one of the stoppers  740 ,  742  attached to the housing  720 . The stiffness of each of the buckled columns  730 ,  731 ,  732 ,  733  is immediately increased and results in a nearly essential nonlinearity similar to that achieved in the NES  600 . Columns have been used in a similar manner to create compact nonlinear springs, but not with the intention of creating a nearly essential nonlinearity. 
   As the broadband force is applied to the housing  720  and/or the primary structure  710 , the broadband force induces energy in the housing  720  and/or the primary structure  710 . As a result, the housing  720  and/or the primary structure  710 . The energy of the housing  720  and/or the primary structure  710  is absorbed by the NES  700 , the stoppers  740 ,  742  and/or the buckled columns  730 ,  731 ,  732 ,  733 . The energy of the primary structure  710  and/or the housing  720  is transferred to the buckled columns  730 ,  731 ,  732 ,  733 . As a result, the energy causes the NES mass  726  to move in the direction of motion  701 . The energy is dissipated from the primary structure  710  and/or the housing  720  via the NES mass  726 , the stoppers  740 ,  742  and/or the buckled columns  730 ,  731 ,  732 ,  733 . Energy transferred to the NES  700  may be ultimately dissipated, for example, through the displacement of air or another surrounding fluid by the motion of the NES mass  726  and/or the buckled columns  730 ,  731 ,  732 ,  733 . 
     FIG. 11  illustrates a nonlinear energy sink  800  (hereinafter “NES  800 ”) in an embodiment of the present invention. Further, the NES  800  has a housing  820 , a mounting flange  822 , guides  824 ,  825  and/or a nonlinear energy sink mass  826  (hereinafter “NES mass  826 ”). The guides  824 ,  825  connect the NES mass  826  to the housing  820  and constrain the NES mass  826  inside of the housing  820 . The NES  800  is attached to a primary structure  810  via the mounting flange  822 . The NES mass  826  may be, for example, a plate, a disk, a cube and/or the like. The NES mass  826  may be made from a material, such as, for example, steel, plastic, fiberglass, wood, cement and/or the like. The present invention should not be deemed as limited to the embodiments of a specific material of the mass  826 . 
   Springs  830 ,  831  are interposed between the NES  826  and the housing  820 . The springs  830 ,  831  may be deformed by a motion of the NES  826 . Each of the springs  830 ,  831  thae the form of a “loop” anchored at two points to two points inside of the housing  820 . Each of the springs  830 ,  831  contact the mass over a region which varies as the NES mass  826  moves. Because the springs  830 ,  831  are a constant length, a free length between the housing  820  and the NES mass  826  changes with displacement of the NES  826 . As a result, a nonlinear stiffness characteristic is produced. An advantage of the NES  800  is that the design is simplicity of construction, achieved at a cost of greater design effort compared to the NES  600 . 
   As the broadband force is applied to the housing  820  and/or the primary structure  810 , the broadband force induces energy in the housing  820  and/or the primary structure  810 . As a result, the housing  820  and/or the primary structure  810 . The energy of the housing  820  and/or the primary structure  810  is absorbed by the NES  800  and/or the springs  830 ,  831 . The energy of the primary structure  810  and/or the housing  820  is transferred to the springs  830 ,  831 . As a result, the energy causes the NES mass  826  to move in the direction of motion  801 . The energy is dissipated from the primary structure  810  and/or the housing  820  via the NES mass  826  and/or the springs  830 ,  831 . Energy transferred to the NES  800  may be ultimately dissipated, for example, through the displacement of air or another surrounding fluid by the motion of the NES mass  826  and/or the springs  830 ,  831 . 
   It should be understood that various changes and modifications to the presently preferred embodiments described herein will be apparent to those skilled in the art. Such changes and modifications may be made without departing from the spirit and scope of the present invention and without diminishing its attendant advantages. It is, therefore, intended that such changes and modifications be covered by the appended claims.