CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims the benefit of U.S. Provisional Application No. 61/621,329, filed Apr. 6, 2012, the disclosure of which is incorporated herein by reference. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    This invention relates in general to a flexible exercise device. In particular, this invention relates to a user-excited, frequency-responsive, structure that provides a force feedback input to exercise muscle groups to improve swing acceleration of sports implements. 
         [0003]    Exercise devices provide various types of force inputs to muscle groups to provide resistance or strength training. Often, these devices use springs or other load/deflection responsive mechanisms to create a force that is resisted by a user to strengthen and tone muscles. The typical excitation of these devices by a user is below the structure&#39;s first natural frequency, relying the spring&#39;s stiffness to provide resistance that is proportional to the deflection of the user. Other exercise devices utilize resilient members, such as flexible rod, loaded in torsion or bending. The resilient members provide resistance as the device is deflected, but the user&#39;s excitation cadence is below the structure&#39;s resonance. 
         [0004]    Other flexible devices are configured to improve the structure and path of a user&#39;s swing relative to the user, the ball, or the ground. These devices are not provided to exercise muscle groups but instead permit a user to develop muscle memory in order to engrain a particular swing path or style. Thus, the excitation frequency of the device is not a consideration in the design of the device or implementation of the workout routine. 
         [0005]    Thus, it would be desirable to provide a muscle development tool that creates strength and muscle tone quickly and easily. It would further be desirable to provide a frequency responsive exercise device that improves the swing speed and acceleration that a user can impart to a sports swing implement, such as a baseball bat, softball bat, tennis racquet, hockey stick, golf club, and the like. 
       SUMMARY OF THE INVENTION 
       [0006]    This invention relates to a flexible exercise device configured to increase muscle tone, speed response, and strength. The flexible exercise device described herein includes a frequency and damping ratio configured for strength training rather than swing tempo training. 
         [0007]    In one aspect of the invention, a flexible exercise device includes a resilient portion connected between a handle and an excitation mass. The resilient member is tuned such that the spring rate of the resilient portion, in combination with the excitation mass, cooperate to provide a natural frequency that is below a user excitation frequency. The flexible exercise device providing a force feedback input into targeted muscle groups that influence a swing speed of a sports implement, such as a baseball bat. 
         [0008]    In one embodiment, a flexible exercise device includes a handle, a spring portion, and an excitation mass configured such that a natural frequency of at least the spring portion and the excitation mass is lower than a user excitation frequency. 
         [0009]    In another embodiment, a flexible exercise device having a spring portion and an excitation mass configured to respond to a user excitation in a post-resonance frequency range such that a therapeutic effect is generated at the end of an oscillatory cycle. The therapeutic effect being a function of the velocity profile of the excitation mass where the excitation mass reverses direction as part of the oscillatory motion of a user forcing function that excites the structure. 
         [0010]    In another embodiment, a flexible exercise device have a spring portion and an excitation mass that cooperate to have a natural frequency above a use excitation frequency. The spring portion is formed from a polymer rod having a length in a range of about 35 inches to about 40 inches and a diameter of about 0.25 inches. The natural frequency is in a range of about 1.7 hertz to about 2.3 hertz and a damping ratio is in a range of about 0.006 to about 0.017. In a variation of this embodiment, the spring portion may include a first resilient element and a second resilient element where the first resilient element is the polymer rod formed from a fiberglass material and the second resilient element is a polymer sleeve coaxially disposed over the first resilient element and having a durometer in a range of about 60 to 80 Shore A. The first and second resilient elements cooperate with the excitation mass to produce a cantilevered first natural frequency of about 2.0 hertz and the excitation mass is about 5.5 ounces. 
         [0011]    Various aspects of this invention will become apparent to those skilled in the art from the following detailed description of the preferred embodiment, when read in light of the accompanying drawings. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0012]      FIG. 1A  is an elevational view of a flexible exercise device. 
           [0013]      FIG. 1B  is a schematic illustration of a cross sectional shape of a resilient portion of the flexible exercise device of  FIG. 1A . 
           [0014]      FIG. 1C  is a schematic illustration of an alternative cross sectional shape of another embodiment of a resilient portion of a flexible exercise device, similar to  FIG. 1A . 
           [0015]      FIG. 2  is a schematic illustration of an embodiment of a spring portion of the flexible exercise device of  FIG. 1 . 
           [0016]      FIG. 3  is a schematic illustration of another embodiment of a spring portion of the flexible exercise device of  FIG. 1 . 
           [0017]      FIG. 4A  is a schematic illustration of an exploded view of an embodiment of a flexible exercise device. 
           [0018]      FIG. 4B  is a schematic illustration of an embodiment of a two stage resilient spring portion of the flexible exercise device of  FIG. 4A . 
           [0019]      FIG. 5  is a schematic illustration of a cantilever-mounted flexible exercise device, showing a deflected test configuration. 
           [0020]      FIG. 6  is a plot of a frequency response function of a flexible exercise device, similar to the device of  FIG. 5 . 
           [0021]      FIG. 7  is a plot of acceleration amplitude vs. time and a vibration decay over time for the tested flexible exercise device, similar to  FIG. 5 . 
           [0022]      FIG. 8  is a graphical estimation of damping ratio of the tested embodiment of a flexible exercise device, similar to  FIG. 5 . 
           [0023]      FIG. 9A  is a table of natural frequency and damping ratio results of four tested embodiments of flexible exercise devices, similar to  FIG. 5 . 
           [0024]      FIG. 9B  is another table, similar to  FIG. 9A , summarizing free vibration testing data. 
           [0025]      FIG. 10  is a schematic illustration, in partial cross section, of an embodiment of a flexible exercise device having a moveable weight. 
           [0026]      FIG. 11A  is a schematic illustration, in partial cross section, of an embodiment of a damped-oscillation moveable weight for a flexible exercise device. 
           [0027]      FIG. 11B  is a schematic illustration, in cross section of another embodiment of a damped-oscillation moveable weight for a flexible exercise device. 
           [0028]      FIG. 12  is a schematic, plan view of a user and the flexible exercise device during use. 
       
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
       [0029]    Referring now to the drawings, there is illustrated in  FIG. 1  an embodiment of a flexible exercise device, shown generally at  10 . The flexible exercise device  10  includes a spring section  12 , a handle  14 , and an excitation mass  16 . The flexible exercise device  10  defines an overall length, L, from a distal end of the handle  14  to a proximal end of the excitation mass  16 . The spring section  12  is shown in  FIGS. 1A and 1B  as a round solid rod having a length, L 1  and a diameter, D. In other embodiments, the spring section  12  may have other cross sectional shapes. In other embodiments, the cross sectional shape of the spring section  12  may be asymmetrical and exhibit two different spring rates relative to the cross section. For example, as shown in  FIG. 1C , the cross sectional shape of a spring section  12   a  may be elliptical and exhibit greater stiffness in the major diameter plane, A and lower stiffness in the minor diameter plane, B. As shown in  FIG. 8 , the plane A is generally oriented parallel to the height of the user. Plane B represents the plane of exercise in which the exercise device oscillates and is oriented at an angle to plane A, such as a 90 degree angle. Typically, the relative angle between planes A and B are based on a comfortable position of the user&#39;s forearms, bent at the elbows, relative to the user&#39;s upper arm. Thus, in one use orientation of the asymmetrical cross section embodiment of the exercise device of  FIG. 1C , the orientation of the spring section  12   a  may be such that the lower stiffness orientation of plane B is in the plane of exercise that is generally parallel to a plane of a user&#39;s shoulders. The higher stiffness orientation of plane A is such that the excitation of the device  10  remains generally in the exercise plane. In alternative embodiments, the planar stiffness differences may be formed into fiber layup orientations rather than asymmetric cross sections, if desired. 
         [0030]    In one specific embodiment, the overall length, L 1  of the spring section  12  may be in the range of 30-40 inches and the diameter, D may be in the range of ⅛ (0.125) inches to ½ (0.5) inches and is dependent on material properties. In another embodiment, L 1  is approximately 38 inches and D is approximately 0.25 inches, though the diameter and length may be varied to achieve a desired stiffness characteristic. The handle  14  has a length, L 2 , and in the illustrated embodiment, L 2  is approximately 10 inches, though other lengths may be provided. An effective spring length of spring section  12  is estimated at approximately 28 inches (i.e. L 1 -L 2 ). This effective spring length, along with the spring material properties, determines the spring stiffness, k, which provides a force input to the user during exercise training, as will be described in detail below. The force input to the user is also a function of the weight of the mass  16  and its position along the spring section relative to the handle  14 , which is generally in the range of 4-6 ounces. In one embodiment, the mass  16  is 5.2 ounces. The weight of the excitation mass  16  may be increased or decreased to change the natural frequency of the exercise device to match the user&#39;s physical requirements. In the illustrated embodiment of  FIG. 1 , the excitation mass  16  is attached to the spring portion  12  by a threaded section  18 . However, the mass  16  may be attached by any other means such as, for example, adhesive bonding, set screws, bolts, jam nuts, and the like. In the illustrated embodiment, the mass  16  has a length L 3  and is attached a fixed distance away from the handle  14 . The mass  16  may be formed of any suitable material and may be covered in a soft medium, such as a foam or neoprene, to minimize damage or injury from incidental contact. The handle  14  may be provided as any grip promoting arrangement such as, for example, a bonded foam sleeve, leather wrapped section, knurled section of the spring  12 , plastic molded section (formed and contoured or straight and smooth). 
         [0031]    In one embodiment, the spring portion  12  may be a molded or extruded polymer material, such as Delrin (e.g. Delrin 570) by DuPont. Alternatively, the spring portion  12  may be made from any material or in any configuration that exhibits a spring rate in the ranges described herein, such as a metal or a fibrous composite material. As shown in  FIG. 2 , a spring section  112  may be formed as a fibrous composite material formed from filaments or fibers, such as fiber glass, aramid fiber, carbon fiber, and the like, in a resin matrix. In the embodiment of  FIG. 2 , the spring section  112  is formed relative to a longitudinal axis, A that establishes a generally 0 (zero) degree axis of orientation. Winding angles of other layers are expressed in degrees relative to this 0 degree longitudinal axis. A fiber orientation of fibers  120  may be generally parallel to axis A. These fibers may be applied to form the spring  112  by any suitable process such as, for example, pultrusion, extrusion, filament winding, weaving, or as a hand lay-up process. A layer of fibers  122  may be formed over the fibers  120  in a relative angular fiber orientation, shown as angle “a,” that is between 90 degrees and approximately 85 degrees relative to axis A. The fibers  122  may also be applied by high volume production processes, such as pultrusion. The fibers  120  are oriented to provide a generally tensile and flexural strength capability. The fibers  122  are applied to provide hoop strength to the longitudinal fibers  120  of the spring structure. The structural stiffness of the spring portion  112  may be varied by apply fibers  124  and  126  at an off-axis winding angle. Such an angle may be, for example, a ±45 degree angle relative to the axis, A. The fibers  124  may be wound of otherwise applied onto surface of the spring layers or a mandrel at an angle, “b” relative to the axis A. In one embodiment, the angle “b” may be approximately 45 degrees to the axis A. It should be understood that other angles may be used to achieve different stiffness properties from the spring portion  112 . For example, as the fiber angle “b” becomes larger, i.e., the fibers  124  are oriented more toward the hoop direction, the fiber properties contribute less to the flexural stiffness while the resin or binding agent properties become more dominant. The fibers  126  may be oriented at an angle “c” that is generally at the same angle as angle “b”, such as 45 degrees to axis A and generally perpendicular to fibers  124 . It should also be understood that fibers  126  may be applied at any desired angle, whether or not the same as angle “b”. 
         [0032]    Referring now to  FIG. 3 , another method may be used to form a spring  212  where one or more sheets  215  of pre-woven fibers impregnated with a resin or binder, i.e., prepreg material, may be rolled or formed into the shape of the spring  212 . The prepreg material  215  may be a woven mat having a first fiber  220  oriented in a first direction and a second fiber  222  oriented in a second direction relative to the first fiber  220 . As shown in  FIG. 3 , the first fibers  220  are generally perpendicular to the second fibers  222 , though other angles between fibers  220  and  222  may be provided. The fibers  220  and  222  are woven together, similar to a woven sheet or cloth, and may be oriented at any desired angle relative to an axis, such as axis A in  FIG. 2 . The sheet  215  is then rolled, typically over a shaped mandrel, in a direction shown by arrow “B”. In the embodiment shown in  FIG. 3  the shape of the spring element is generally conical (and either hollow or solid), but other shapes and profiles may be used. The spring  212  may then be processed using heat, vacuum, pressure, and/or mechanical compression to conform the material to the desired shaped mandrel and provide a curing of the resin. 
         [0033]    Referring now to  FIGS. 4A and 4B , there is illustrated an exploded view of an embodiment of a flexible exercise device  300 . The flexible exercise device  300  includes a first flexible or resilient element  312  that connects a handle  314  to an excitation mass  316 . The handle  314  has a bore  314   a  extending at least partially down the center length thereof. The first resilient element  312  may be solid or hollow, straight or tapered, and formed of any suitable material. Disposed over the first resilient element  312  is a second resilient element  318 , shown as a polymer tubular sheath. The second resilient element  318  may be formed as a tubular plastic sleeve member having an outer diameter pressed inside the handle bore  314   a  and an inner diameter that accepts the outer diameter of the first resilient element  312 . The first and second resilient elements  312  and  318  are illustrated as being disposed in a coaxial orientation. The second element  318  may be configured as a series of discrete elements disposed about the circumference of the first element  312 , if desired. The first and second resilient elements  312  and  318  may be positioned such that each element cooperates as a parallel-functioning multiple spring portion, where each is subjected to generally the same deflection. 
         [0034]    The first and second resilient elements  312  and  318 , and also the handle  314 , may be secured together by any means, such as adhesive bonding, chemical welding, ultrasonic welding, mechanical fasteners, or a mechanical press fit, or any combination thereof. The second element  318  may be provided in different durometers and associated press fits with the first element  312 . The second resilient element may be, for example, a plastic or polymer tube such as may be formed from nylon, polyester, polyvinyl chloride, polyurethane and the like. In one embodiment, the second element  318  has a durometer in the range of approximately 60-80 Shore A, and more specifically in a range of 72-74 Shore A. In another embodiment the durometer of the second element  318  may be in a range of 80-100 Shore A. 
         [0035]    In certain embodiments, adjusting the amount of press fit or interference fit between the first and second resilient elements  312  and  318  changes both the resulting spring stiffness and the amount of damping. As the press fit between the first and second resilient elements  312  and  318  is reduced, a controlled amount of relative movement occurs at the interface therebetween. The relative movement, in the presence of a compressive load, provides a frictional force component resulting in damping applied to the device  300 . Also, as the press fit relationship is reduced or eliminated, the resilient elements are subjected to different deflections, due in part to the different lengths of each element during deflection (such a deflection is illustrated in  FIG. 5 ), the contribution of spring rate times deflection differs. 
         [0036]    In an alternative embodiment, the first and second resilient elements  312  and  318  may be bonded together by an adhesive, a chemical or thermal weld, or a mechanical connection. As the fit relationship between the first and second resilient elements  312  and  318  is increased, the relative differences in deflection are reduced such that both elements are subjected to the same deflection and the length of the second resilient element  318  is stretched more compared to the first resilient element  318 . This deflection/strain configuration results in less frictional damping, more material damping and more spring rate contribution of the second resilient element  318  to the overall response of the structure. In another embodiment of a flexible exercise device, a plastic sleeve, similar to the second resilient element  318  may be limited to the interface region between the handle  314  and the first resilient element  312 . The plastic tubing may be nylon, polyester, polyvinyl chloride, polyurethane and the like. The plastic sleeve, in this configuration acts as a shock absorber between the handle and the resilient element to provide cushioning to the user&#39;s hands. 
         [0037]    Referring now to  FIG. 5 , there is illustrated a deflection mode shape of the flexible exercise device  10 . The deflected mode shape is illustrated as a deflected cantilever beam approximating a half order mode shape exhibiting a single node point at the handle  14 . Tests conducted using various embodiments of the illustrated embodiment fix the handle  14  of the flexible exercise device  10  relative to the spring portion  12 . In one test condition, the handle  14  is held in a generally horizontal attitude. The spring portion  12  and the excitation mass  16  extend in a generally horizontal direction and have a static deflection of the spring portion  12  caused by the excitation mass  16 . Alternatively, the flexible exercise device  10  may be fixtured at any other relative orientation, such as a vertical orientation. For tests conducted using a free vibration excitation, the spring portion  12  and mass  16  are deflected an amount of delta 1 , Δ 1 , and released, allowing the device  10  to vibrate freely. The spring deflection may be within the plane of orientation or at a relative angle thereto. As shown in  FIG. 7 , the spring portion  12  and mass  16  oscillate in a generally sinusoidal pattern with generally decreasing amplitudes of vibration. Alternatively, the spring may be deflected by an amount of delta 2 , Δ 2 . The deflection amounts delta 1 , Δ 1  and delta 2 , Δ 2 , may be the same or different. Referring back to the embodiment of the flexible exercise device  10  of  FIG. 1 , described above, the deflected amount Δ 1  may be between 18 and 22 inches. More particularly, for a deflection of 20 inches, a load of approximately 6 lbs. is required. The resultant spring rate (k=F/Δ) of the spring  12  is about 0.3 lbs. per inch. Other spring rates may be used in combination with other mass weights to provide a desired natural frequency characteristic. 
         [0038]    The various tested embodiments were also evaluated using a force-transducing hammer and an accelerometer as inputs to a Fast Fourier Transformer (FFT) vibration analyzer. A frequency plot vs. amplitude from the testing is shown in  FIG. 6  and a vibration amplitude sweep showing the time rate of decay is shown in  FIG. 7 .  FIG. 8  is an enlarged plot of the frequency response function from one of the test trials. The graph of  FIG. 8  is marked to determine damping ratio. The various embodiments of the flexible exercise device are configured to exhibit a lightly damped, natural frequency of about 2 Hz. As shown in the Tables of  FIGS. 9A and 9B , four embodiments of the flexible exercise device exhibit natural frequencies and damping ratios that are in a range of 1.5 Hz. to 3 Hz and about 0.006 to about 0.018, respectively. Table  9 A identifies the four test specimens and test results for a first battery of tests using both a free vibration excitation and impact hammer excited response testing. Free vibration excitation uses an accelerometer and a single force input to cause deflection and oscillation of the structure, resulting in measurement of the vibratory forcing function. Impact response testing uses a force transducing hammer that inputs a known forcing function and an accelerometer measures the oscillatory output to determine the natural frequency response of the structure. 
         [0039]    The tabulated test results of specimens tested are generally configured as described hereafter. Specimen 1 is configured with parallel spring arrangement having a first resilient element that is an approximately 0.25 inch diameter pultruded fiberglass rod having a Young&#39;s modulus of about 6×10 6  psi. One example of such a pultruded rod is an Extren® rod manufactured by Strongwell Corporation, Bristol, Va. A second resilient element that is part of the parallel spring arrangement is a coaxially oriented polyurethane sleeve is fitted over the rod to form the two spring portion, similar to the embodiment of  FIGS. 4A and 4B . The polyurethane sleeve has a durometer of approximately 73 Shore A. The excitation mass is approximately 4 ounces and the specimen length, L is about 38.5 inches. Specimen 2 is configured as a parallel spring arrangement with a first resilient element of 0.25 inch diameter pultruded fiberglass rod, similar to Specimen 1. The second spring element is a polyurethane sleeve having a durometer of 85 Shore A and an excitation mass of about 5.0 oz. The length L of Specimen 2 is approximately 37.5 inches. Specimen 3 is also configured as a parallel spring arrangement having the same first resilient element of a 0.25 inch O.D. fiberglass rod. The second resilient element is a urethane sleeve having a durometer of 64 Shore A. The excitation mass is about 4.0 oz. and the length, L is about 37.5 inches. Specimen 4 is configured as a single spring portion configured as a pultruded fiberglass rod without an outer sleeve. The handle length, L2 of four tested specimens are generally equal. 
         [0040]    The embodiments of the flexible exercise device described above are tunable to accommodate the force input and muscle development requirements of various users, from the novice to the professional athlete. By varying the length of the spring portion 12, the spring rate and natural frequency may be varied. As one of the length of the spring portion increases or the weight of the excitation mass increases, the natural frequency decreases. As the diameter of the spring portion increases, the stiffness of the spring portion increases as does the natural frequency of the flexible exercise device. An increase in spring stiffness may be counteracted by increasing the weight proportionally so that the natural frequency remains the same. The higher the stiffness and the mass increase, the exercise energy levels go up, thus becoming tailored for a stronger user. 
         [0041]    Referring now to  FIG. 10 , there is illustrated an embodiment of a flexible exercise device, shown generally at  400 , having a spring portion  412  that terminates in a handle  414  fixed at a distal end of the spring portion  412 . The spring portion  412  includes a moveable weight  416  generally positioned at a proximal end of the spring portion  412 . The moveable weight  416  is adjustable along a length L 4  of the spring portion  412 . The moveable weight  416  may be moved by a distance ΔL such that the center of the weight is generally positioned at a distance L offset  from the proximal end of the spring portion  412 . In one embodiment, the moveable weight  416  includes a bore  418  that is configured and sized relative to the outer diameter of the spring portion  412  to permit the weight to be axially displaced along the spring portion, yet sized so as not to exhibit relative movement with the spring portion in the plane of oscillation when put in use. The bore  418  may include a mechanical retaining device, such as a set screw, ball and detents, collet and jamb nut, or other suitable structure to fix the desired position of the weight relative to the spring portion  412 . In one embodiment, the mass of weight  416  may be a fixed value or, alternatively, the weight can be adjustable such that the mass value is increased or decreased to adjust the natural frequency. The resulting movement of the weight  416  and/or altering of the mass provides the ability to adjust the mass spring system and configure the exercise device  400  as a stiffer or more compliant structure. Generally, the further away the weight is moved from the handle  414 , the lower the natural frequency response. Also, the larger the mass value at a given location, the lower the natural frequency becomes. 
         [0042]    Referring now to  FIGS. 11A and 11B , there are illustrated alternative weight structures that include a moving weight with damping. As shown in  FIG. 11A , a moveable weight  500  includes a fluid filled chamber  502  and a moving piston  504  disposed within the chamber. The piston  504  is fixed to a proximal end of a spring portion  512  that is similar to the various spring portion embodiments described herein. The piston  504  may include one or more passages  506  that extend through the length of the piston  504  and may also include one or more seals, such as o-rings  508 , though such is not required. The passages  506  are sized to permit a fluid  510  to pass through as the weight is slowly moved relative to the piston  504 . As the weight  500  and the piston  502  move at higher velocities relative to each other, the fluid is sheared as it flows through the passages  506 , producing a damping effect similar to a shock absorber. The fluid  510  may be any suitable martial, such as a water, latex, or oil based material, a thixotropic paste, a compressed gas, or other shear-able medium. The fluid  510  may be provided on one side or both sides of the piston  504 . The fluid on one side may also be a compressible and expansible medium. The fluid may also responsive to an energy field, such as a magneto-rheological fluid, electro-rheological fluid, piezo-ceramic material, and the like. 
         [0043]      FIG. 11B  illustrates a similar, alternative embodiment of a moveable weight  600 . The moveable weight  600  includes a chamber  602  and a piston  604  disposed within the chamber  602 . The piston  604  may include at least one passage  606  and at least one seal  608 , similar to those described above. The chamber  602  is filed, at least on one side of the piston  604 , with a fluid  610 . The piston  604  is attached to a spring portion  612 , similar to the spring elements described above. A resilient member  614 , illustrated as a coil spring, is disposed on the other side of the piston  604 . The resilient member  614  is illustrated as a compression coil spring that acts against the force of weight  600  caused by centripetal acceleration during oscillation. 
         [0044]    Referring now to  FIG. 12 , there is illustrated the flexible exercise device  10  being oscillated by a user in accordance with a general method of use to develop specific muscle groups in the hands, wrists, forearms, triceps, shoulders and chest areas. In operation, the user grips the handle, either with one hand or both. As shown in  FIG. 12 , the user grips the handle  14  with both hands. The user articulates the wrists back and forth, as shown by deflections  20   a  and  20   b,  in an out-of-phase motion. The out-of-phase motion of the user&#39;s wrists compresses certain muscle groups in one hand while extending or stretching the same muscle groups of the other hand. The excitation oscillations of the user are inputted at a rate above the natural frequency of the spring/mass system of the flexible exercise device  10 , such as about 2.5 Hz. The excitation oscillations  20   a  and  20   b  cause the spring portion  12  to oscillate within the plane B with a handle deflection  22 . 
         [0045]    The energy of the excitation oscillations of the user transmits down the length of the spring portion  12  and oscillates the excitation mass  16  with a mass deflection  24 . The excitations of the mass  16 , operating in a post-first resonance condition, transmit a multiplied force input into the handle  14 , which is then resisted and counteracted by the user. The resistance force is counteracted as the excitation mass decelerates to a zero velocity and changes direction. Thus, the flexible spring portion amplifies at least one of deflection and the acceleration of the excitation mass that is imparted by the user. 
         [0046]    The various embodiments of the flexible exercise device produce a therapeutic effect that builds muscle tone and structure. The operation of the flexible exercise device in a post-resonance condition (i.e., above the first natural frequency of the cantilever structure) creates both isolation of forces that can cause muscle strain and a therapeutic resistance that promotes muscle mass and tone. The post-resonance operation reduces the forces on muscles during the swinging motion as the force transmissibility of the stiffness input of the spring portion is reduced. The therapeutic resistance that excites the muscles happens at the ends of travel where the velocity profile of the excitation mass slows to a sub-resonance condition and passes through a zero velocity, at least in the exercise plane of excitation. There is a response lag between when the user&#39;s hands change direction of the excitation imparted to the device and when the excitation mass goes to a zero velocity and changes direction. The resistance of this rebounding force is supplied by the user&#39;s forearm, triceps, and upper torso muscles. The rate of force input, because of the post-resonance operation, causes the muscle groups to be exercised rapidly. Though the deflections of the muscles are relatively low, compared to similar muscle exercises using free weights or weight resistance machines, the excitation frequency is significantly higher. Thus, the energy input of the muscle groups has the benefits of a low impact regiment like those of isometric exercising, yet rapid repetition rates greater than those of resistance machines. This combination of low muscle deflection and high repetition rate creates a low impact workout where the targeted muscle groups are loaded and relaxed in rapid succession. 
         [0047]    The principle and mode of operation of this invention have been explained and illustrated in its preferred embodiment. However, it must be understood that this invention may be practiced otherwise than as specifically explained and illustrated without departing from its spirit or scope.
A flexible exercise device includes a resilient portion connected between a handle and an excitation mass. The resilient member is tuned such that the spring rate of the resilient portion, in combination with the excitation mass, cooperate to provide a natural frequency that is below a user excitation frequency. The flexible exercise device provides a force feedback input into targeted muscle groups that influence a swing speed of a sports implement, such as a baseball bat.