Abstract:
Vibration damping apparatus, systems, objects including such apparatus and systems, and vibration damping methods. The vibration damping involves amplifying a vibration-induced displacement and damping the amplified displacement.

Description:
BACKGROUND  
       [0001]    1. Field 
         [0002]    The present inventions relate generally to vibration damping. 
         [0003]    2. Related Art 
         [0004]    Vibration damping materials are employed in a wide variety of mechanical systems to damp vibrations that can degrade performance of the systems. The present inventors have determined conventional vibration damping materials are susceptible to improvement. For example, low stiffness materials are commonly used for vibration damping because their flexible lattices are inherently better at dissipating energy. In some instances, however, devices must be securely held in place despite the fact that the devices are sensitive to, or are the source of, vibrations. Launch vehicle adaptors, motor mounts, and high precision moving devices are examples of such devices. Low stiffness materials are less than optimal here because they permit movement, as are conventional high stiffness materials because they have relatively poor damping characteristics. 
       SUMMARY  
       [0005]    A vibration damping apparatus in accordance with one embodiment of a present invention includes a displacement conversion device configured to convert displacement in a first direction into displacement in a second direction, a motion amplifier responsive to displacement in the second direction, and a motion damper operably connected to the motion amplifier. The present inventions also include systems and objects with such vibration damping apparatus. 
         [0006]    A vibration damping method in accordance with one embodiment of a present invention includes the steps of converting a vibration-induced displacement in a first direction into a displacement in a second direction, amplifying the displacement in the second direction to create an amplified displacement, and damping the amplified displacement. 
         [0007]    There are a variety of advantages associated with such a vibration damping apparatus, systems, objects and methods. For example, the present vibration damping apparatus may be configured such that it is relatively stiff in the direction of the vibration forces (e.g., the first direction), while providing better damping characteristics than have been heretofore associated with materials and structures having the same stiffness in the direction of the vibration forces. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS  
         [0008]    Detailed description of preferred embodiments of the inventions will be made with reference to the accompanying drawings. 
           [0009]      FIG. 1  is a schematic diagram of a damping apparatus in accordance with one embodiment of a present invention. 
           [0010]      FIG. 2  is a schematic diagram of a damping apparatus in accordance with one embodiment of a present invention. 
           [0011]      FIG. 2A  is a graph showing force v. displacement curves for various damping coefficients. 
           [0012]      FIG. 3  is a perspective view of a damping apparatus in accordance with one embodiment of a present invention. 
           [0013]      FIG. 4  is a side view of a portion of the damping apparatus illustrated in  FIG. 3 . 
           [0014]      FIGS. 4A and 4B  are sections views of an exemplary fluid-filled machine. 
           [0015]      FIG. 5  is a plan view of a spring that may be employed in at least some damping apparatus in accordance the present inventions. 
           [0016]      FIG. 6  is a side view of the spring illustrated in  FIG. 5 . 
           [0017]      FIG. 7  is a force v. displacement curve for an exemplary spring. 
           [0018]      FIG. 8  is a side view of a portion of damping apparatus in accordance with one embodiment of a present invention. 
           [0019]      FIG. 9  is a side view of a damping apparatus in accordance with one embodiment of a present invention. 
           [0020]      FIG. 10  is a side view of a portion of the damping apparatus illustrated in  FIG. 9 . 
           [0021]      FIG. 11  is a perspective view of a damping system in accordance with one embodiment of a present invention. 
           [0022]      FIG. 12  is a perspective view of a damping system in accordance with one embodiment of a present invention. 
           [0023]      FIG. 13  is a perspective view of a damping system in accordance with one embodiment of a present invention. 
           [0024]      FIG. 14  is a perspective view of a damping system in accordance with one embodiment of a present invention. 
           [0025]      FIG. 15  is a perspective view of a damping system in accordance with one embodiment of a present invention. 
           [0026]      FIG. 16  is a perspective view of a payload attach fitting in accordance with one embodiment of a present invention. 
           [0027]      FIG. 17  is a side view of a frame system in accordance with one embodiment of a present invention. 
           [0028]      FIG. 18  is a side view of a shear wall in accordance with one embodiment of a present invention. 
           [0029]      FIG. 19  is a side view of a boat hull in accordance with one embodiment of a present invention. 
           [0030]      FIG. 20  is a side view of an airplane wing in accordance with one embodiment of a present invention. 
       
    
    
     DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS  
       [0031]    The following is a detailed description of the best presently known modes of carrying out the inventions. This description is not to be taken in a limiting sense, but is made merely for the purpose of illustrating the general principles of the inventions. 
         [0032]    A vibration damping apparatus (or “damping apparatus”) in accordance with one embodiment of a present invention is generally represented by reference numeral  100  in  FIGS. 1 and 2 . The exemplary damping apparatus  100  includes a displacement conversion device  102 , a motion amplifier (or “displacement amplifier”)  104 , and a motion damper (or “displacement damper”)  106 , each of which is discussed in greater detail below. Briefly, the displacement conversion device  102  converts displacement thereof along a first axis A 1  (or “in a first direction”) caused by vibrational force F VIB  into displacement along a second axis A 2  (or “in a second direction”). The first and second axes A 1  and A 2  (or “different directions”) may be transverse (including perpendicular as shown) or otherwise non-coaxial. The period of the back and forth vibration-induced displacement along each axis is the same. In at least some instances, the magnitude of the displacement along the second axis A 2  (i.e., the displacement distance Δ 2 ) is greater than the magnitude of the displacement along the first axis A 1  (i.e., the displacement distance Δ 1 ) and, in view of the fact that the periods are the same, the velocity of the displacement along the second axis is greater than the velocity of the displacement along the first axis. 
         [0033]    The motion amplifier  104  includes an interface member  108  that also moves along the second axis A 2  and whose motion is damped by the motion damper  106 . The displacement of the interface member  108  along the second axis A 2  is, relative to the displacement of the displacement conversion device along the first axis A 1 , amplified both in magnitude and velocity. In particular, the motion amplifier  104  creates displacement (i.e., the displacement distance Δ 3 ) at the interface member  108  that is greater than the displacement distance Δ 2  in response to the displacement of portions of the displacement conversion device  102  along the second axis A 2 . The period of the back and forth vibration-induced displacement of the interface member  108  is the same as that of the displacement conversion device  102  along axes A 1  and A 2 . As such, the velocity of the interface member  108  is greater than the velocity of the displacement conversion device  102  along axis A 1  and along axis A 2 . Thus, in the illustrated implementation, the vibration-induced displacement (and velocity) that occurs along axis A 1  is amplified, and then the amplified displacement (and velocity) is itself amplified. 
         [0034]    Given that damping force is equal to the damping coefficient multiplied by the velocity to the damped object (F d =cv), damping the twice amplified motion at the interface member  108  results in far more effective damping of vibrations applied to damping apparatus  100  than would be the case if the motion damper  106  was acting directly on movement of the displacement conversion device  102  along the first axis A 1  or the second axis A 2 . As such, the present vibration damping apparatus  100  may be configured such that it effectively damps vibrations despite being relatively stiff in the direction of the vibration forces. 
         [0035]    The exemplary motion amplifier  104  includes a positive spring  110  (i.e., a spring with a positive spring constant K 1 ) and a negative spring  112  (i.e., a spring with a negative spring constant K 2 ). As used herein, a “negative spring” is spring which has a stress-strain (or load-displacement) curve with a negative slope. In some instances, the negative slope will be over only a portion of the stress-strain (or load-displacement) curve. Here, the spring may be selected, preloaded and/or physically restrained such that it operates within the region of negative slope. As is explained below, the spring constants K 1  and K 2  are close to, but not exactly, equal and opposite in value. The absolute value (or “magnitude”) of the negative spring constant K 2  is slightly less than the absolute value (or “magnitude”) of the positive spring constant K 1 . As used herein, “slightly less” is 1% to 10% less. 
         [0036]    By way of background, it should be noted that when positive and negative springs which have spring constants that are equal in absolute value, i.e., have equal and opposite spring constants, are combined in parallel the resulting spring constant is zero. Conversely, when two springs of equal and opposite spring constant are combined in series, the resulting spring constant is infinity. If an outside force is applied to the positive/negative spring series, the overall length of the spring series remains constant, but the interface between the positive and negative springs will move as the positive spring compresses and the negative spring expands in response to the force. The inventors herein have determined that this phenomenon is magnified when there is slight difference between the absolute values of the spring constants in the positive/negative spring series. The magnitude of the interface displacement is inversely proportional to the percent difference between the absolute values of the positive and negative spring constants. Put another way, motion amplification=K 1 /(IK 1 I−IK 2 I). If, for example, the percent difference between the spring constant absolute values is 2%, with a slightly positive overall spring constant, then the displacement at the interface will be 50 times the combined displacement at the ends of the positive/negative spring series. The present inventors have also determined that the positive/negative spring series, which is slightly positive in overall spring constant, is relatively unstable. As such, stabilization of a positive/negative spring series may be required. 
         [0037]    Referring more specifically to  FIG. 2 , the configuration of the displacement conversion device  102  in the illustrated embodiment is such that it defines a spring constant K 3  (or “stiffness”) in the direction of the second axis A 2  and a spring constant K 4  (or “stiffness”) in the direction of the first axis A 1 . Spring constant K 3  is a positive spring constant that stabilizes the motion amplifier  104  that includes positive and negative springs  110  and  112 . Spring constant K 4  is a positive spring constant that defines the stiffness of the damping apparatus  100  in the direction of the vibrational force F VIB . Spring constant K 3  is greater than the effective spring constant of the positive/negative spring series K EFF , where K EFF =(K 1 K 2 )(/K 1 +K 2 ). For example, K EFF  may range from 0.5 K 3  to 0.9 K 3 . Spring constant K 4 , on the other hand, should be high enough, given the expected magnitude of the vibrational force F VIB  associated with the intended application, that the displacement along the first axis A 1  (i.e., the displacement distance Δ 1 ) is minimized, but not so high that the displacement is non-existent, as it is the displacement along the first axis A 1  that is amplified along axis A 2  for damping. 
         [0038]    For any particular damping apparatus, there is an optimal value for the damping coefficient c of the motion damper  106 , i.e., a value that results in the most damping for that system.  FIG. 2A  shows the force versus displacement curves for otherwise identical damping apparatuses with different damping coefficients, including the optimum damping coefficient c for that apparatus, damping coefficients that are less (0.1 c and 0.01 c) and damping coefficients greater (10 c and “infinite,” i.e., high enough that the interface member  108  does not move). The most damping occurs where the area within the associated hysteresis curve is the greatest.  FIG. 2A  also shows that damping apparatus in accordance with the present inventions can be configured (or “tuned”) for a particular application. Some applications may, for example, call for greater stiffness and this can be achieved by increasing the damping coefficient with the understanding that the damping will be less than optimal. 
         [0039]    Turning to  FIGS. 3 and 4 , the exemplary damping apparatus  200  illustrated therein operates in the manner described above and includes structures that are functionally similar to those illustrated in  FIGS. 1 and 2 . To that end, the exemplary damping apparatus  200  includes a displacement conversion device  202 , a motion amplifier  204  with an interface  208 , and a motion damper  206 . The displacement conversion device  202  converts (and amplifies) displacement thereof along the first axis A 1  caused by vibrational force F VIB  into greater displacement (and velocity) along the second axis A 2 , the motion amplifier  204  further amplifies the already amplified displacement (and velocity) along the second axis A 2 , and the twice amplified motion is then damped by the motion damper  206  at the motion amplifier interface  208 . 
         [0040]    The exemplary displacement conversion device  202  includes a pair of arcuate members  214  and  216 , with inner surfaces  214   a  and  216   a  and outer surfaces  214   b  and  216   b,  which are secured to one another at their longitudinal ends. The arcuate members  214  and  216  function like leaf springs, and the length, width, thickness, arc and materials may be selected such that, when combined, the displacement conversion device  202  will have the desired spring constant K 3  (along axis A 2 ) and spring constant K 4  (along axis A 1 ). The arcuate members  214  and  216  may be formed from any material, or combination of materials (e.g., a multi-layer composite), that is appropriate for the intended application. Suitable materials include, but are not limited to, plastics, metals such as aluminum and steel, and composites such as fiber reinforced resin composites. The arcuate members  214  and  216  may also be identical (as shown) or may be different in one or more aspects such as, for example, their curvature, thickness or material. 
         [0041]    In addition to the interface  208 , the exemplary motion amplifier  204  includes a positive spring  210  and a negative spring  212 . The exemplary springs  210  and  212  are positioned on the outer surfaces  214   b  and  216   b  of the conversion device arcuate members  214  and  216  in the illustrated embodiment. The outer surfaces  214   b  and  216   b  support the outer rims of the springs  210  and  212  and include indentations under the springs that allow the springs to compress. The interface  208  mounts the springs  210  and  212  to the arcuate members  214  and  216  and, to that end, includes a rod  218  and a pair of end caps  220  and  222  that are secured to the rod. The rod  218  extends through the springs  210  and  212  and the arcuate members  214  and  216 , and the end caps  220  and  222  engage the springs. So arranged, the illustrated interface  208  and springs  210  and  212  are connected in series. i.e., spring-interface-spring. The rod  218  is also movable relative to the arcuate members  214  and  216 . The connectors between the rod  218  and end caps  220  and  222  may be threaded connectors so that the distance between the end caps  220  and  222  can be adjusted to, for example, add and/or adjust a preload on the springs  210  and  212 . Such preloading is discussed below with reference to  FIGS. 5-7 . The interface  208  also includes a damper engagement device  224 , which may be a flat plate as it is in the illustrated embodiment, that is secured to (and therefore moves with) the rod  218 . 
         [0042]    Turning to the motion damper  206 , a wide variety of dampers may be employed. Such dampers may include, but are not limited to, sheets of high damping materials such as rubber, silicone, and foams (e.g., closed cell foams), frictional dampers, and mechanical apparatus such as shock absorbers and other hydraulic and pneumatic dampers. In the illustrated implementation, and although the present inventions are not so limited, the motion damper  206  includes a plurality of hollow hourglass-shaped machines  226  which are filled with regions of incompressible fluid that are separated from one another by a compressible material or gas bubble. Briefly, and referring to  FIGS. 4A and 4B , the exemplary machines  226  may include a pair of opposing planar walls  228  that are connected by transverse walls  230 , which are semi-rigid or flexible, and have a curved shape. End walls  231  ( FIG. 4 ) are located at each longitudinal end. The walls  228  and  230  together define a generally hourglass-shaped internal space  232  that is filled with one or more volumes of an incompressible fluid  234  (e.g., water, silicone oil, non-Newtonian fluids such as shear thinning fluids and shear thickening fluids, hydraulic fluid, and magneto rheological fluids) and one or more volumes of a compressible fluid  236  (e.g., air or another inert gas). When the planar walls  228  are subjected to compressive forces, the transverse walls  230  may bend and bow inwards into the internal space  232  and the compressible fluid  236  will be compressed to accommodate encroachment of the incompressible fluid  234 . In other implementations, the entire internal space  232  may be filled with a gas such as air. Alternatively, the entire internal space  232  may be filled with an incompressible fluid and the end walls  231  may be configured to bulge outwardly. Additional information concerning such dampers is disclosed in U.S. Pat. No. 6,830,793, which is incorporated herein by reference in its entirety. 
         [0043]    The machines  226  are secured to arcuate member inner surfaces  214   a  and  216   a  and to the damper engagement device  224 . As the rod  218  and damper engagement device  224  moves back and forth, some of the machines  226  will be in tension and some of the machines will be in compression. As such, the machines  226  will not add or subtract to the positive spring constant K 1 . If on the other hand the machines were only located on one side of the damper engagement device  224 , then the spring constant of the machines would form part of the positive spring constant K 1 . 
         [0044]    The positive and negative springs  210  and  212  are not limited to any particular springs, so long as the springs have the requisite positive or negative spring constants. Referring to  FIGS. 5 and 6 , and although the present inventions are not so limited, the exemplary negative spring  212  may be a conical disc  238  with stress-relieving cutouts  240  and  242  on the inner and outer diameters. Negative springs of this type are available, for example, from Associated Spring under the trade name Clover® Dome. Positive spring  210  is also a conical disc spring with stress-relieving cutouts on the inner and outer diameters in the illustrated embodiment. Positive springs of this type are also available from Associated Spring under the trade name Clover® Dome. As illustrated for example in  FIG. 7 , there are some instances where a negative spring will have a negative spring constant over only a portion of its load v. displacement curve. Here, the negative spring may be preloaded by, for example, selecting component dimensions, adding a spacer (note spacer  244  in  FIG. 8 ) and/or adjusting the end caps  220  and  222  in such a manner that the negative spring is compressed into the portion of the load v. displacement curve that has a negative slope. It may also be desirable to prevent the negative spring from being compressed beyond the portion of the load v. displacement curve that has a negative slope. This may be accomplished by, for example, mechanical stops that prevent compression past a particular point or an overall damping apparatus configuration where, for the intended application, one would not expect displacement beyond the portion of the curve with the negative slope. 
         [0045]    In some implementations, positive spring and motion damper functionalities may be performed by a common structure. For example, fluid fill machines such as those described above with reference to  FIGS. 3 and 4  may be used to provide positive spring and motion damper functionalities. One example of a damping apparatus with such an arrangement is generally represented by reference numeral  200   a  in  FIG. 8 . Damping apparatus  200   a  is substantially similar to damping apparatus  200  in form and function and similar elements are represented by similar reference numerals. For example, damping apparatus  200   a  includes a displacement conversion device  202  with a pair of arcuate members  214  and  216 , and a motion amplifier  204   a  with a negative spring  212 , a rod  218 , and end caps  220  and  222 . Here, however, the damper engagement device  224  of the interface  208   a  is not located between the arcuate members  214  and  216  and, instead, abuts the end cap  220 . A plurality of fluid-filled hourglass-shaped machines  226 , which function as both a positive spring and a damper, are located between the arcuate member outer surface  214   b  and the damper engagement device  224 . In particular, and referring to  FIGS. 4A and 4B , the curved transverse walls  230  function as a positive spring and the compressible fluid  236  functions as a damper. It should also be noted that an annular spacer  244  is located between the negative spring  212  and the arcuate member  216  to support the outer rim of the negative spring away from the arcuate member so that the negative spring can compress. 
         [0046]    Another exemplary damping apparatus is generally represented by reference numeral  300  in  FIGS. 9 and 10 . Damping apparatus  300  is substantially similar to damping apparatus  200  and damping apparatus  200   a  in various aspects of their form and function, and similar elements are represented by similar reference numerals. For example, damping apparatus  300  includes a displacement conversion device  302 , a motion amplifier  304 , and a motion damper  306 . There is also a plurality of the above-described fluid-filled machines  226 , which have dual positive spring and damping functionality. Here, however, the displacement conversion device  302  has a half-leaf configuration with a planar member  314  and an arcuate member  316 . Such a configuration is better suited for vibration based displacement that tends to bend the damping apparatus  300  in the manner and direction identified by the arrows in  FIG. 9 . Such bending is about axis A 3 , which is perpendicular to both axis A 1  and axis A 2 , and results in some displacement in the directions of the first and second axes A 1  and A 2 . The motion amplifier  304  includes positive and negative springs and an interface, and creates an amplified displacement (as compared to the displacement along axis A 2 ) at the interface. Here, the positive spring is defined by the transverse walls  230  ( FIG. 4A ) of the machines  226  that are located against and secured to the planar member inner surface  314   a  and a damper engagement device  324 , and the negative spring  312  is carried on the arcuate member outer surface  316   b.  The exemplary motion amplifier interface  308  has a rod  318  that extends through and is movable relative to the arcuate member  316 , a single end cap  322  that secures the negative spring  312 , and the damper engagement device  324  that is secured to (and therefore moves with) the rod  318 . Turning to the motion damper  306 , the exemplary motion damper includes a pair of dampers  306   a  that act directly on the displacement conversion device  302  and are incorporated into K 3 , and a damper  306   b  that acts on the amplified motion associated with the motion amplifier  304 . In the illustrated implementation, the dampers  306   a  and  306   b  are each defined by the compressible fluid  236  ( FIG. 4B ) within the machines  226 . Here too, the damper engagement device  324  is a flat plate. In those instances where it is desirable to provide a flat surface for the machines  226  in dampers  306   a,  a shim  338  with curved and flat surfaces may be provided. 
         [0047]    The present vibration damping apparatus may be incorporated into (or used in combination with) a wide variety of systems and objects, as is described below with reference to  FIGS. 11-20 . The systems illustrated in  FIGS. 11-20  are presented for purposes of example only, and the present inventions are not limited to such systems and objects. 
         [0048]    A structural plate, such as the structural plate  400  illustrated in  FIG. 11 , is one example of a vibration damping system that includes a plurality of the present vibration damping apparatuses. The exemplary structural plate  400  includes a plurality of individual vibration damping apparatuses  200   b  arranged in row and columns, i.e., end-to-end and side-to-side. Three rows of five damping apparatuses  200   b  are shown in the illustrated embodiment. Systems with a single or column may also be provided. The damping apparatuses  200   b  are essentially identical to the damping apparatus  200  in structure and function and similar elements are used to represent similar reference numerals. Here, however, the displacement conversion device  202   b  in each apparatus includes ends tabs  215  and  217  where the arcuate members  214  and  216  are connected to one another. The displacement conversion devices  202   b  are secured adjacent displacement conversion devices. Welding, adhesive, mechanical fasteners, or any other suitable instrumentality may be used to connect the damping apparatuses  200   b  to one another. In other implementations, the arcuate member  214  (and/or the arcuate member  216 ) for two, more than two, or all of the damping apparatuses  200   b  may be formed from a single piece of material that is sized and shaped so as to define a plurality of arcuate members. Other vibration damping apparatuses, such as damping apparatuses  200 ,  200   a  and  300 , may be employed in a structural plate in place of the damping apparatuses  200   b.  Also, although the exemplary structural plate  400  employs a plurality of identical damping apparatuses, other structural plates may employ damping apparatuses that differ in one or more aspects. Also, a plurality of structural plates may be combined, e.g., into a T-shape or an I-beam. 
         [0049]    A structural tube, such as the structural tube  410  illustrated in  FIG. 12 , is another example of a vibration damping system that includes a plurality of the present vibration damping apparatuses. The exemplary structural tube  410  includes a plurality of individual vibration damping apparatuses  200  positioned about the longitudinal axis LA. The first axis A 1  of each damping apparatus  200  is parallel to the longitudinal axis LA. The longitudinal ends of the damping apparatuses  200  are fixed relative to one another and, in the illustrated embodiment, are secured to anchor rings  412  and  414 . The structural tube  410  is intended to be oriented such that the longitudinal axis LA, and the first axes A 1 , are parallel to the vibration forces F VIB . Welding, adhesive, mechanical fasteners, or any other suitable instrumentality may be used to connect the damping apparatuses  200  to the anchor rings  412  and  414 . Other vibration damping apparatuses, such as damping apparatuses  200   a,    200   b  and  300 , may be employed in a structural tube in place of the damping apparatuses  200 . Structural tubes may also be formed from columns of vibration damping apparatuses that are secured to one another end to end. Also, although the exemplary structural tube  410  employs a plurality of identical damping apparatuses, other structural tubes may employ damping apparatuses that differ in one or more aspects. A bicycle seat post is one example of device that may be formed by the structural tube  410 . 
         [0050]    A structural tube, such as the structural tube  420  illustrated in  FIG. 13 , is another example of a vibration damping system that includes a plurality of the present vibration damping apparatuses. The exemplary structural tube  420  includes a plurality of individual vibration damping apparatuses  200   c  positioned about the longitudinal axis LA. The first axis A 1  of each damping apparatus  200   c  is tangential to the longitudinal axis LA. So configured, vibrational forces F VIB  that are perpendicular to the longitudinal axis LA cause displacement along the first axes A 1  of the damping apparatuses  200   c.  The damping apparatuses  200   c  are essentially identical to the damping apparatus  200   b  and similar elements are used to represent similar reference numerals. Here, however, the displacement conversion devices  202   c  are relatively long in the direction parallel to the longitudinal axis LA, and include relatively long arcuate members  214   c  and  216   c  and end tabs  215   c  and  217   c.  Each damping apparatus  200   c  also includes a plurality of motion amplifiers  204  and associated motion dampers  206 . Welding, adhesive, mechanical fasteners, or any other suitable instrumentality may be used to connect the damping apparatuses  200   c  to one another. Other vibration damping apparatuses, such as damping apparatuses  200 ,  200   a,    200   b  and  300 , may be employed in a structural tube in place of the damping apparatuses  200   c.  Also, although the exemplary structural tube  420  employs a plurality of identical damping apparatuses, other structural tubes may employ damping apparatuses that differ in one or more aspects. 
         [0051]    Other examples of vibration damping systems are grid systems such as the isogrid system  430  illustrated in  FIG. 14 . The exemplary isogrid system  430  includes a plate  432  and a plurality of triangular trusses  434  that are formed from damping apparatuses  200 . So arranged, the triangular trusses  434  perform a stiffening function as well as the above described vibration damping function. Vibrational forces acting on the plate  432  are transferred to the longitudinal ends of the damping apparatuses  200 , in the direction of axis A 1 , by way of the anchor posts  436  that define the corners of the triangular trusses  434 . The damping apparatuses  200  are connected to anchor posts  436 , but are not directly connected to the plate  432 . The plate  432  may be part of the structure whose vibrations are being damped, and the plate may be planar or curved. By way of example, by not limitation, a curved plate (or a plurality of connected curved plates) may define a cylindrical shape so that the structure may be employed in a rocket or other launch vehicle. Although the trusses  434  in the exemplary system  430  are equilateral triangles, the present systems are not so limited. Other patterns and arrangements including, but not limited to, squares, octagons and other geometric shapes, parallels, diagonals and combinations thereof (e.g., z-shapes), may be employed. Other vibration damping apparatuses, such as damping apparatuses  200   a - c  and  300 , may be employed in the grid system in place of the damping apparatuses  200 . Also, although the exemplary isogrid system  430  employs a plurality of identical damping apparatuses, other systems may employ damping apparatuses that differ in one or more aspects. 
         [0052]    Another exemplary grid system is the isogrid system  440  illustrated in  FIG. 15 . The exemplary isogrid system  440  includes a plate  442  and a plurality of triangular trusses  444  that are formed from damping apparatuses  300   a.  The damping apparatuses  300   a  are similar to apparatus  300  in form and function, and similar elements are represented by similar reference numerals. For example, each damping apparatus  300   a  includes a displacement conversion device  302   a  with an arcuate member  316 , a motion amplifier  304   a  with a negative spring  312 , a motion damper  306   a  with a plurality of fluid-filled machines  226  that perform the positive spring and damping functions in the manner described above. A damper engagement device  324   a  is secured to the motion amplifier interface rod (not shown). In contrast to damping apparatus  300 , however, the damping apparatus  300   a  omits the planar member  314  ( FIG. 9 ) and is configured to be mounted onto the surface of the underlying structure (e.g., the inner surface of the plate  442 ). The ends of the arcuate members  316  are secured to the plate  442  by anchor posts  446  that define the corners of the triangular trusses  444 . The plate  442  may be planar or curved. By way of example, by not limitation, a curved plate (or a plurality of connected curved plates) may define a cylindrical shape so that the structure may be employed in a rocket or other launch vehicle. The machines  226  are located between the arcuate member  316  and the damper engagement device  324   a,  and between the damper engagement device and the plate  442 . In other implementations, the machines  226  will be on only one side of the damper engagement device  324   a.  Although the trusses  444  in the exemplary system  440  are equilateral triangles, the present systems are not so limited. Other patterns and arrangements including, but not limited to, squares, octagons and other geometric shapes, parallels, diagonals and combinations thereof (e.g., z-shapes), may be employed. Other vibration damping apparatuses, such as damping apparatuses  200 - 200   c  and  300 , may be employed in the grid system in place of the damping apparatuses  300   a.  Here, the anchor posts  446  would also provide a spacing function to suspend the damping apparatuses relative to the plate  442 . Also, although the exemplary isogrid system  440  employs a plurality of identical damping apparatuses, other systems may employ damping apparatuses that differ in one or more aspects. 
         [0053]    Other examples of vibration damping systems are strut systems, such as the strut system  452  in the payload attach fitting (or “forward adapter”)  450  illustrated in  FIG. 16 . The strut system  452  includes a plurality of struts  454 , and each strut includes a plurality of damping apparatuses  200   b.  Other aspects of the payload attach fitting  450  include the payload interface  456  and the spacecraft interface ring  458  which are located at opposite ends of the strut system  452 . The struts  454  are connected to anchors  460  and  462  on the payload interface  456  and spacecraft interface ring  458 . The vehicle fuel tank  466  is also visible in  FIG. 16 . 
         [0054]    Other examples of vibration damping systems are braced-frame systems, such as the braced-frame system  470  illustrated in  FIG. 17 . The exemplary braced-frame system  470  includes horizontal and vertical frame members  472  and  474 , and braces  476 . Vibration damping apparatus, such as damping apparatus  200 , may be positioned in series with the braces  476  (e.g., at the longitudinal ends of the braces) and/or in parallel with the braces  476  (e.g., structural plates  400  that are coextensive with braces), and/or in series or parallel with the frame members  472  and  474 . The damping apparatuses  200  may be configured to dampen low-amplitude vibrations before the remainder of the braced-frame system  470  acts on large amplitude motions. 
         [0055]    Turning to  FIG. 18 , the present vibration damping systems also have application in reinforced concrete shear walls. The exemplary shear wall  480  illustrated in  FIG. 18  includes a concrete wall  482  mounted on a frame  484 . Structural plates  400  may be embedded within the concrete. The structural plates  400  which may be oriented horizontally, vertically or at an angle in between (as shown), are stiffer than the concrete and therefore attract (and damp) vibrations. 
         [0056]    Another exemplary application is boat hulls. The exemplary boat hull  490  illustrated in  FIG. 19  incorporates one or more of the present vibration damping apparatus and/or systems. To that end, the exemplary boat hull  490  consists of an internal network of frames (not shown), that extend from side to side and that run the length of the boat, that are covered by an outer shell  492  (e.g., fiberglass or metal). Other boat hulls consist solely of the outer shell. The frames and/or shell may be formed from one or more of the above-described vibration damping apparatus and/or systems. The apparatus and systems illustrated in  FIGS. 9 ,  10  and  15  are especially applicable to the outer shell of a boat hull. 
         [0057]    The exemplary airplane wing  500  illustrated in  FIG. 20  also incorporates one or more of the present vibration damping apparatus and/or systems. The exemplary wing  500  may include an external skin  502  and internal structures (not shown) such as ribs, stringers and spars. The external skin  502  and/or the internal structures may be formed from one or more of the above-described vibration damping apparatus and/or systems. The apparatus and systems illustrated in  FIGS. 9 ,  10  and  15  are especially applicable to the external skin of an airplane wing. 
         [0058]    Although the present inventions have been described in terms of the preferred embodiments above, numerous modifications and/or additions to the above-described preferred embodiments would be readily apparent to one skilled in the art. It is intended that the scope of the present inventions extends to all such modifications and/or additions.