Abstract:
A method for doubling an input motion of a first mechanism including: providing the first mechanism having a first link, a second link rotatably connected to the first link; a first output which undergoes a motion resulting from a motion of the first link, the first output being operatively connected to the first link through at least the second link; driving the first link from a first position to a second position, wherein a singular position of the first and second links occurs between the first and second positions; cascading the first mechanism with a second mechanism; and inputting the second mechanism with the output to double the output and quadruple the input.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is a divisional application of U.S. patent application Ser. No. 10/932,580 filed on Sep. 2, 2004, which claims the benefit of earlier filed provisional patent application 60/499,444 filed Sep. 2, 2003, entitled “On The Existence Of Special Cases Of Input Speed Doubling Linkage Mechanisms,” the contents of each of which are incorporated herein by its reference. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    1. Field of the Invention 
         [0003]    The present invention relates generally to linkages, and more particularly, to frequency doubling planar and spatial linkage mechanisms. 
         [0004]    2. Prior Art 
         [0005]    A number of investigators have studied the harmonic content of closed-loop linkage mechanisms with rigid links and have developed direct analysis and synthesis methods based on the harmonic content of the output motion. It has been shown that if the motion of the input link of a linkage mechanism were periodic with a fundamental frequency ω, the output motion would also be a periodic motion with the same fundamental frequency ω. However, since linkage mechanisms commonly have a nonlinear input-output motion relationship, the output motion would also contain harmonics of the input motion harmonics. For example, if the input link of a four-bar or slider-crank linkage mechanism turns angular velocity of ω, or if it oscillates with a simple harmonic motion with a frequency ω, the output link would undergo a periodic motion with the fundamental frequency ω and a number of its harmonics. In only a few special cases, e.g., in four-bar parallelogram mechanisms, the input-output relationship is linear and therefore the output motion has the same number of harmonics as the input motion. 
         [0006]    The fact that the input and the output motions have to have identical fundamental frequencies can also be explained from the fact that in general, during one cycle of input motion, the output has to complete its motion cycle, therefore should have the same fundamental frequency as the input. For example, in a four-bar crank-crank (crank-rocker) mechanism, one full turn of the input link can only result in one continuous turn (rocking motion) of the output link. In addition, since during one full turn of the input link the coupler and output link chain has to stay within one of their two configurations, the rocker can only make a single continuous back and forth motion between its two extreme positions. This is obviously also the case for rocking input link motion, i.e., if the input link undergoes one continuous back and forth motion, then output link undergoes one back and forth motion. This argument is obviously true for any linkage mechanism. 
         [0007]    It can therefore be said that as it is known to date and in general, for a continuous full rotation or a continuous rocking motion of the input link of a linkage mechanism, the output link can only undergo a continuous rotation or a continuous rocking motion. The only exceptions that have been discovered to date are the Galloway type of mechanisms. In these crank-crank type of planar and spatial linkage mechanisms, two turns of the input link results in one full turn of the output link. It has been shown that such motions are possible only in certain special cases. In such cases, one full cycle of the input link rotation occurs in one configuration (branch) and the second cycle in a second configuration of the linkage chain that starts from the output link and extend to the moving joint of the input link. For example, in the Galloway (or deltoid) mechanism, during one full rotation of the input link, the open-loop output and coupler link chain is in one configuration, and during the second full turn of the input link, the chain is in its second configuration. 
       SUMMARY OF THE INVENTION 
       [0008]    Therefore, it is an objective of the present invention to overcome the deficiencies of the prior art mechanisms. 
         [0009]    Accordingly, a mechanism is provided. The mechanism comprising: a first link; a second link rotatably connected to the first link; a first output which undergoes a motion resulting from a motion of the first link, the first output being operatively connected to the first link through at least the second link; and an input actuator for driving the first link from a first position to a second position, wherein a singular position of the first and second links occurs between the first and second positions. 
         [0010]    The input actuator can drive the first link in a rocking motion. The input actuator can drive the first link in a full rotation motion. 
         [0011]    The mechanism can further comprise a third link rotatable connected to the second link at one end and operatively connected to the first output at another end. 
         [0012]    The input actuator can be a motor. The input actuator can be a link from another mechanism. 
         [0013]    The mechanism can further comprise: a third link operatively connected to the first output; a fourth link rotatably connected to the third link; and a second output which undergoes a motion resulting from a motion of the third link, the second output being operatively connected to the third link through at least the fourth link; wherein the first output drives the third link from a third position to a fourth position, wherein a singular position of the third and fourth links occurs between the third and fourth positions. 
         [0014]    The output can be configured to drive a shaker. The output can be configured to drive a mixer. The output can be configured to drive a crusher. 
         [0015]    Also provide is a device comprising: a first member; a second member rotatable connected to the first member; an output which undergoes a motion resulting from a motion of the first member, the output being operatively connected to the first member through at least the second member; and an input actuator for driving the first member from a first position to a second position, wherein a singular position of the first and second members occurs between the first and second positions. 
         [0016]    Still provided is a device comprising: a first member; a second member rotatable connected to the first member; an output which undergoes a motion resulting from a motion of the first member, the output being operatively connected to the first member through at least the second member; and an input actuator for driving the first member through a range of motion which includes a singular position of the first and second members. 
         [0017]    Still further provided is a device for suppressing an input motion. The device comprising: an input which undergoes a motion; an output at which the motion is suppressed; a first linkage operatively connecting the input and output, the first linkage comprising a first link rotatably connected to the output at a first portion and a second link rotatably connected to a second portion of the first link at a third portion and to a damper at a fourth portion, the damper being operatively connected to the input; wherein the first and second links are driven through their singular position by the motion. 
         [0018]    The device can be a suspension of a vehicle, where the input is one or more wheels of the vehicle and the output is a chassis of the vehicle. 
         [0019]    The device can further comprise: a second linkage operatively connecting the input and output, the second linkage comprising a third link rotatably connected to the output at a fifth portion and a fourth link rotatably connected to a sixth portion of the third link at a seventh portion and to a damper at a eighth portion, the damper being operatively connected to the input; wherein the first, second, third and fourth links are driven through their singular position by the motion. 
         [0020]    Still further provided is a method for doubling an input motion of a first mechanism. The method comprising: providing the first mechanism having a first link, a second link rotatably connected to the first link; a first output which undergoes a motion resulting from a motion of the first link, the first output being operatively connected to the first link through at least the second link; and driving the first link from a first position to a second position, wherein a singular position of the first and second links occurs between the first and second positions. 
         [0021]    The method can further comprise: cascading the first mechanism with a second mechanism; and inputting the second mechanism with the output to double the output and quadruple the input. 
         [0022]    Still yet provided is a method for doubling an input motion of a first mechanism. The method comprising: providing the first mechanism having a first link, a second link rotatably connected to the first link; a first output which undergoes a motion resulting from a motion of the first link, the first output being operatively connected to the first link through at least the second link; and driving the first link through a range of motion which includes a singular position of the first and second members. 
         [0023]    The method can further comprise: cascading the first mechanism with a second mechanism; and inputting the second mechanism with the output to double the output and quadruple the input. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0024]    These and other features, aspects, and advantages of the apparatus and methods of the present invention will become better understood with regard to the following description, appended claims, and accompanying drawings where: 
           [0025]      FIG. 1  illustrates a schematic of a slider-crank linkage mechanism having an input motion as is known in the prior art. 
           [0026]      FIG. 2  illustrates a schematic of an embodiment of a slider-crank linkage mechanism of the present invention. 
           [0027]      FIG. 3  illustrates an embodiment of a schematic of a four-bar linkage mechanism of the present invention. 
           [0028]      FIG. 4  illustrates an embodiment of a schematic of a crank-rocker type of mechanism of the present invention. 
           [0029]      FIG. 5  illustrates an embodiment of a schematic of the output of the mechanism of  FIG. 4  used as an input to a second motion-doubling mechanism. 
           [0030]      FIG. 6  illustrates a plot of the input and the resulting motion and fundamental frequency-doubled output motion for an example of the mechanism of  FIG. 3 . 
           [0031]      FIG. 7  illustrates a schematic of an embodiment of an isolation system of the present invention. 
       
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
       [0032]    The present invention discloses special classes of planar and spatial linkage mechanisms in which for a continuous full rotation or continuous rocking motion of the input link, the output link undergoes two continuous rocking motions. Such mechanisms are hereinafter referred to as “motion-doubling” linkage mechanisms. 
         [0033]    In a special case of such mechanisms, for periodic motions of the input link with a fundamental frequency ω, the output motion is periodic but with a fundamental frequency of 2ω. This mechanism is hereinafter referred to as the “fundamental frequency-doubling” linkage mechanism. 
         [0034]    The motion-doubling linkage mechanisms can be cascaded to provide further doubling of the output rocking motion. Such mechanisms may be cascaded with other appropriate linkage mechanisms to obtain crank-rocker or crank-crank type of mechanisms. Furthermore, a doubling of a full rotation or rocking motion into a rocking motion can be converted into a full rotation (which is doubled) by use of a piston/crankshaft arrangement (where the output rocker is the piston which turns a crankshaft). Such piston/crankshaft arrangements for converting a rocking motion (an oscillation) into a rotating motion are well known in the art. 
         [0035]    In addition, a special class of linkage mechanisms is presented in which for a full continuous rocking motion of the input link, the coupler link undergoes two continuous rocking motions. Such mechanisms are referred to as coupler motion-doubling linkage mechanisms. Similarly, the output rocking motion can be converted to a full rotation motion, such as with a piston/crankshaft arrangement. 
         [0036]    Such motion-doubling mechanisms have practical applications, particularly when higher output or coupler speeds are desired, since higher output or coupler motions can be achieved with lower input speeds. 
         [0037]    In addition, such mechanisms also generally have force transmission and dynamics advantages over regular mechanisms designed that could be used to achieve similar output or coupler speeds, in the sense that their links and joints are subject to lower dynamics forces and can therefore be designed with lighter weight components and are subject to less vibration related problems. In addition, with by reducing the dynamics forces and the mass of the various components of the linkage mechanism, the actuating motor that is required to drive the mechanism becomes smaller and its dynamics response requirement is greatly reduced, which almost always translates into less expensive and lighter weight actuating motors. 
         [0038]    The conditions for the existence of output and coupler motion-doubling linkage mechanisms and fundamental frequency-doubling linkage mechanisms are and their mode of operation is described below. 
         [0039]    Consider the slider-crank linkage mechanism shown in  FIG. 1 . The input link  102  with the length a makes an angle θ with the X-axis of the fixed XY coordinate system. The input at O A  can be any input known in the art, generally referred to as an actuator  101 , such as a motor. The coupler link  104  has a length b. The position of the slider block  106  along the X-axis is shown by s. If the input link  102  at position O A A undergoes a periodic motion with a fundamental frequency ω, e.g., if the motion of the input is the simple harmonic motion 
         [0000]      θ=θ 0 +θ 1  sin(ω t )  (1)
 
         [0000]    where θ 1  is the amplitude of the input link oscillation about the position θ 0 . The output motion s is periodic with the same fundamental frequency ω, and a certain number of its harmonics with significant amplitudes, i.e. 
         [0000]    
       
         
           
             
               
                 
                   s 
                   = 
                   
                     
                       s 
                       0 
                     
                     + 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         
                           s 
                           i 
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 ω 
                                  
                                 
                                     
                                 
                                  
                                 t 
                               
                               + 
                               
                                 φ 
                                 i 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where n is the number of harmonics with significant amplitudes, s 0  is a constant, and s i , i=0, 1, . . . , n is the constant amplitude and φ i  is the phase of the ith harmonic of the output motion. 
         [0040]    The links  102 ,  104  can be of any structural configuration known in the art. Furthermore, the connection between the links  102 ,  104  as well as the connection between the coupler link  104  and the slider block  106  are pivoting (rotating) joints  108 ,  110  as are known in the art. The slider block  106  can be a mass or a portion of the coupler link  104 , which is confined along the x-axis. As discussed below, the output can also be associated with another device, another linkage, a bracket for holding an object (e.g., a paint can), or an end effector (e.g., a tool for crushing solid objects). 
         [0041]    Let a cycle of the harmonic motion (1) of the input link  102  start from the position O A A (solid lines), continue to the position O A A′ (dashed lines) during the first half of the cycle of motion, and bring the input link back to its starting position O A A during the second half of the cycle of motion. During this motion, the output slider block  106  moves from its starting position B to the position B′ during the first half of the cycle of motion, and moves back to the position B during the second half of the cycle of motion, i.e., during each cycle of motion, the output slider block  106  undergoes one cycle of back and forth motion. 
         [0042]    Referring now to  FIG. 2 , a linkage  200  is illustrated therein similar in construction to that shown in  FIG. 1  but driven by an input actuator in a different manner to achieve a different and novel result at the output thereof. Consider the case in which the harmonic motion (1) of the input link  102  starts from the position O A A (solid lines),  FIG. 2 , continues to the position O A A″ (dotted lines), which is symmetrically positioned with respect to the X axis, during the first half of the cycle of motion, and brings the input link  102  back to its starting position O A A during the second half of the cycle of motion. During this motion, the output slider block  106  moves from the position B to the position B′ as the input link  102  moves from the position O A A to the position O A A′, where the input link  102  and the coupler link  104  are collinear, i.e., are in their singular position. As the input link  102  motion continues from the position O A A′ to the position O A A″, the output slider block  106  moves back to its starting position B. The back and forth motion of the output slider block  106  is repeated as the input link  102  rotates back from its O A A″ position to its starting position O A A. Thus, during one back and forth cycle of the input link  102  motion, the output slider block  106  undergoes two back and forth motions. In this special case of symmetrical motion of the input link  102  about the singular position of the input link  102  and coupler link  104 , the two back and forth motions of the output slider block  106  are identical, each constituting a simple harmonic motion with the fundamental frequency 2ω. The motion of the output slider block  106 , equation (2), is thereby reduced to 
         [0000]    
       
         
           
             
               
                 
                   s 
                   = 
                   
                     
                       s 
                       0 
                     
                     + 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         m 
                       
                        
                       
                         
                           s 
                           i 
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 2 
                                  
                                  
                                  
                                 
                                     
                                 
                                  
                                 ω 
                                  
                                 
                                     
                                 
                                  
                                 t 
                               
                               + 
                               
                                 φ 
                                 i 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where m is the number of harmonics with significant amplitudes. 
         [0043]    The output motion is therefore doubled, i.e., the output motion is periodic and its fundamental frequency has been doubled. It can also be said that one back and forth motion of the input link  102  results in two back and forth motion of the output slider block  106 . The above motion doubling occurs for all input motions as long as the motion during both forward and return half cycles of the input link  102  motion are identical except in their direction. As discussed above, the rocking output motion can be converted to a full rotation motion, such as with the use of a piston/crankshaft arrangement as is known in the art. 
         [0044]    In the general case of non-symmetrical motion of the input link  102  about its singular (θ=0) position, the two back and forth motions of the output slider block  106  are not identical, and the motion of the output slider block  106  is still described by equation (2), i.e., the fundamental frequency of the output motion is still ω and is not doubled. However, during one back and forth cycle of input link  102  motion, the output slider block  106  still undergoes two back and forth motions, i.e., the output motion is doubled. 
         [0045]    The reason why two back and forth motions of the output slider block  106  can be achieved for each single back and forth motion of the input link  102  is as follows. The input link  102  and coupler link  104  chain can place the output slider block  106  in a specified position s within their reachable space with two different configurations or branches, noting such configurations or branches always have to appear in pairs. When the back and forth motion of the input link  102  is only one of the two configurations of the chain ( FIG. 1 ), the output slider block  106  can only undergo one back and forth motion, since the functions describing such motions are one to one. Thus, the only way that a single back and forth motion of the input link  102  could result in two back and forth motions of the output slider block  106  is when one of the latter motions occurs in one configuration and the second motion in the other configuration of the input link  102  and coupler link  104  chain as is shown in  FIG. 2 . 
         [0046]    In general, the two back and forth motions of the output slider block  106  are not identical, and together constitute one cycle of a periodic function with the fundamental frequency ω of the input motion as described by equation (2). However, when the input motion is symmetrical with respect to the singular position of the input link  102  and the coupler link  104  chain, the two back and forth motions of the output slider block  106  become identical, each constituting a simple harmonic motion with the fundamental frequency 2ω, as described by equation (3). 
         [0047]    Similar input motion doubling occurs in all linkage mechanisms when the input link crosses its singular position with the next (coupler like) link during its back and forth (rocking) motion. As a result, the output link undergoes one “back and forth” (rocking) motion in one configuration and a second rocking motion in the other configuration of the input and coupler link chain. For example, such a motion is illustrated in  FIG. 3  for a four-bar linkage mechanism  300  in which a second coupler link (or output link)  302  is pivotally coupled with the first coupler link  104  at pivot joint  304  at one end and pivotally coupled with an output at pivot joint  308 . Any output device (including another linkage mechanism) known in the art can be coupled to the pivot joint  308  (output). 
         [0048]    Here, during one cycle of the input link  102  motion, the input link  102  starts its motion from the position OA, pass through the singular position of the input link  102  and coupler link  104  OA′ and up to the position OA″, and continuously returns to its starting position OA. Similarly, if the two rocking motions of the output link  302  are identical, the fundamental frequency of the output link  302  motion is doubled. The two rocking motions of the output link  302  are identical when the motion of the input link  102  in each of the two configurations of the input link  102  and coupler link  104  chain are identical, i.e., the motion from the position OA′ to the position OA and back is identical to the motion from the position OA′ to the position OA″ and back. 
         [0049]    In the above two examples illustrated in  FIGS. 2 and 3 , the input link  102  undergoes one rocking motion, crossing the singular position of the input link  102  and coupler link  104  chains during its motion. Such singular position crossings are essential to allow for one rocking motion of the output in one configuration of the input link  102  and coupler link  104  chain and another in the other configuration of the input link  102  and coupler link  104  chain. Such a pattern of singular position crossings is obviously not possible if the input link  102  undergoes a full and continuous rotation, i.e., by crank-rocker or crank-crank type of linkage mechanisms. 
         [0050]    The aforementioned rocking motion of the input link may be, however, generated by another crank-rocker type of mechanism, such as the one shown in  FIG. 4 . In the mechanism  400  of  FIG. 4 , a first linkage  401  consists of links  102 ,  104 , and  402  coupled by rotating joints  403  and  405 . An input actuator  101  drives the input link  102  through a full rotation which results in a rocking motion output at pivoting joint  407 . Link  402  is a three-sided member that serves as an input to a second linkage  404 . The second linkage includes a first link  402   a , which is a portion of link  402  of the first linkage. Link  402   a  is rotatably coupled to link  409  through joint  410 . Link  409  is rotatably coupled to an output link  406  through joint  411  at one end. Another end of output link  406  is rotatably coupled to an output at  408 . As shown in  FIG. 5 , the output of the first mechanism  401  drives the input of the second mechanism  404  through a singular position (shown in solid lines) of link  402   a  and link  409 . Therefore, the output at  408  of the second mechanism is doubled as discussed above. Thus, the combined mechanism  400  of  FIG. 5  is input with a full rotation motion at link  102  and outputs with a rocking motion at  408  having a doubled frequency. 
         [0051]    As shown in  FIG. 5 , motion-doubling mechanisms may be cascaded to quadruple the input motion. For example, the output of the mechanism  400  shown in  FIG. 4  may be used as an input to a second motion-doubling mechanism  500  to further double the input motion at link  102  to obtain a quadrupled output motion at  502 . Mechanism  400  is used to input mechanism  500  which consists of linkage  406  (which now consists of a three-sided linkage member having sides  406   a - c ). Linkage member  406  is rotatably connected to links  504  and  506  through pivoting joints  508  and  510 . When links  406   b  and  504  are driven through their singular position, the output at  502  is doubled with regard to the in put at  408  (which is doubled with regard to input  407 ) resulting in a net effect of quadrupling the input. 
         [0052]    This process of motion doubling may continue to further double the output motion, and in theory there is no limit to this doubling process, but in practice, the output motion generally keeps getting smaller by each motion doubling process. 
         [0053]    An example is provided next for a motion and fundamental frequency-doubling four-bar linkage mechanism. Consider the four-bar linkage mechanism shown in  FIG. 3 . Let the link lengths be a=3.5 cm, b=6.5 cm, c=7.5 cm and d=12 cm. The input motion is considered to be a simple harmonic motion given by 
         [0000]      θ=θ 0 +30 cos(ω t )  (4)
 
         [0000]    where θ 0  is the input angle at the singular position of the input and coupler links and ω is the fundamental frequency of the input motion. 
         [0054]    With the aforementioned link lengths, the angle θ 0  is readily determined to be 38.52 deg. Since the input motion, equation (4), is symmetric about the singular position of the input and coupler link chain, the fundamental frequency of the output motion is doubled and the output link undergoes two rocking motions during each cycle of the input motion. For a fundamental frequency of ω=6 rad/sec, the plot of the input and the resulting motion and fundamental frequency-doubled output motion are shown in  FIG. 6 . 
         [0055]    The disclosed motion and fundamental frequency doubling plane and spatial linkage mechanisms may be coupled to other mechanisms to achieve the desired higher speed motions with slower input actuator (e.g., a motor). The disclosed classes of mechanisms have a wide range of applications, particularly in higher speed machinery and devices where dynamics and vibration become problematic, limit the performance, or make the machinery or device expensive and/or heavy. The following are a number of specific applications for which the disclosed motion and fundamental frequency doubling mechanisms are of significant advantage. 
         [0056]    In a first example, the output link motion is doubled, preferably with the doubled fundamental frequency also doubled. The output motion is then used to drive a shaker (used for example, for sorting, sieving, staining, or the like), a mixer (used for example, for mixing paint, chemicals, various fluids, or other types of materials), or a crusher (such as machinery used to crush various solid materials). 
         [0057]    In another example, the motion of a coupler link of the mechanism is doubled. The output motion is then used to drive a shaker (used for example, for sorting, sieving, staining, or the like), a mixer (used for example, for mixing paint, chemicals, various fluids, or other types of materials, or a crusher (such as machinery used to crush various solid materials). 
         [0058]    In yet another example, the motion-doubling characteristic of the mechanisms of the present invention may be used to construct shock and vibration isolation and suspension systems. For example, passive suspension mechanisms may be constructed in which dampers (or spring-damper units) undergo two cycles for each cycle of input oscillation. Such passive suspensions can also be designed to provide one cycle of damper (or damper-spring units) undergo one cycle of output motion with each cycle of input oscillation when the amplitude of the output oscillation is small. The dampers (or spring-damper units) then undergo two cycles of oscillations when the amplitude of the output oscillation becomes large. The latter mechanisms have the advantage of providing “soft” suspensions as long as the amplitude of the resulting oscillation is small. However, if the amplitude of oscillation becomes large, they would rapidly reduce the amplitude of oscillation by doubling the motion of the dampers (or spring-damper units). 
         [0059]    The schematic of such an isolation mechanism, as used to isolate an oscillating device is shown in  FIG. 7 . Such an isolation system can also be used as a car suspension, in which case, the mechanism is positioned between each wheel axle and the chassis. 
         [0060]    In  FIG. 7 , an oscillating mass  600  with its direction of oscillation is shown. Here two motion-doubling mechanisms  602  (each consisting of a first link  604 , connected to the oscillating mass  600  by a pivoting joint  606  on one end and to a second link  612  on the other end) are used. The first link  604  is connected to the second link  612  by a pivoting joint  614  on one end and to a horizontal spring-damper unit  608  on another end. In the example of  FIG. 7 , the second link  612  is confined to slide in the horizontal direction by collar  616 . The input disturbances are considered to coming from the ground  610  (or base). If the resulting amplitude of oscillation of the mass  600  is small, i.e., during its oscillation, the first and second links  604 ,  612  do not line up along the horizontal line (H) (their singular position), such as the amplitude  618 . In this case, during each oscillation of the mass  600 , the spring and damper units  608  (including unit  608   a  discussed below) undergo one cycle of (back and forth) motion. When the amplitude of oscillation of the mass  600  becomes large, i.e., when during one cycle of oscillation the first and second links  604 ,  612  pass through their singular position, such as the amplitude  620 , then the horizontally positioned spring and damper units  608  undergo two cycles of back and forth motion. As a result, a significantly larger amount of energy is taken out of the system, thereby the oscillations of the mass  600  is reduced (damped) at significantly higher rates. It is noticed that such a significant change in the rate of damping is achieved in a totally passive and automatic manner as the amplitude of oscillation is increased beyond a desired level (as defined by the rest or initial angular position of the links). Another advantage of such shock or vibration isolation or suspension mechanisms is that the spring and damper units  608  attached to the links generate essentially symmetrical loads on the support structure (shown as ground  610  in  FIG. 7 ), thereby making it easier to support as internal forces with minimal dynamics implications. 
         [0061]    A further spring and damper unit  608   a  can be added in the vertical direction and coupled to the mass  600  by a vertical link  604   a  to further stabilize the system. As discussed above, the ground (or the wheel of an automobile)  610  can serve as an input to the system where the output is a damped mass  600  (which can be the chassis of the automobile), however, the mass  600  can also serve as the input to the system where the output is at the ground (e.g., for damping the vibrations of a machine). 
         [0062]    While there has been shown and described what is considered to be preferred embodiments of the invention, it will, of course, be understood that various modifications and changes in form or detail could readily be made without departing from the spirit of the invention. It is therefore intended that the invention be not limited to the exact forms described and illustrated, but should be constructed to cover all modifications that may fall within the scope of the appended claims.