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Fig 2 The wave form observed when a person steps onto the force plate in. Fig 1 The homemade force plate A thin sheet of rubber was glued to the a crouching position then stands up straight then steps off the force plate. outer surfaces of both plates to prevent slipping, gardless of whether one stands in the middle of the plate or. on one corner provided all four piezos are closely matched. in sensitivity, the output of such a disk saturates with a load greater than. about 0 5 kg The piezos used in the force plate were manu III STANDING AND JUMPING WAVE FORMS. factured for sonar applications cost 25 each and were de. signed to operate without saturation in the high pressure en If one stands with both feet on a force plate it will register. vironment at the bottom of the ocean 5 They were made from a force F5M g If the center of mass CM is then lowered. a hard PZT ceramic in the form of square blocks each by bending the knees the force does not remain equal to Mg. 28 mm328 mm35 mm with a silvered electrode covering Instead the force decreases F M g and then increases. each of the two large surfaces The output voltage of a piezo F M g before settling back to Mg Alternatively if one. element is given by V5Q C where C is the capacitance of steps onto the force platform in a crouching position and. the element Q5d 33F is the charge induced by a force F then stands up straight the result shown in Fig 2 is obtained. applied in a direction perpendicular to the electrode surfaces In raising the CM the CM starts with zero speed accelerates. and d 33 is the relevant piezo coefficient 340310212 C N for to finite speed then decelerates to a new resting position. the material chosen Each piezo had a capacitance C During this maneuver the plate registers a force F given by. F2M g5M a where a is the acceleration of the CM verti. 51 9560 01 nF and generated an output voltage of 174 mV. cally upward,per Newton, Figure 3 shows the wave form observed when jumping off. The capacitance of the assembly was 7 80 nF but it was. the floor and dropping onto the force plate from a height of a. artificially increased to 0 57 mF by connecting an external. few cm landing with both feet simultaneously The force. 0 56 mF capacitor as shown in Fig 1 The output signal. rises rapidly to a value significantly larger than Mg The. from the assembly was monitored using a 33 MV resistor in. magnitude of the impact force can be reduced by allowing. series with a 1 MV input impedance digital storage oscillo. the knees to bend more on contact or increased by keeping. scope to increase the RC time constant to 19 s The output. the legs straight The CM has a negative velocity at contact. signal was therefore reduced in amplitude by a factor of 73. decreasing rapidly to zero with a slight positive velocity. by the external capacitor and by an additional factor of 34. overshoot due to flexure of the knees The initial acceleration. due to the series resistor to give an overall sensitivity of 17 5 is therefore large and positive in a direction vertically up. mV per kN the force on each piezo being reduced by a. ward so F M g initially This is a good example where a. factor of 4 since four piezos were used to share the load To. deceleration in one direction can usefully be interpreted as an. monitor the force wave form on a longer time scale and or at. acceleration in the opposite direction The magnitude of the. higher sensitivity force plates are normally connected to a. force is easily calculated using estimates of the initial veloc. charge amplifier This was not necessary for the experiments. ity and the time taken to come to rest Figure 4 shows the. described below since a force was applied to the plate for. wave form that results when jumping off the platform from a. only a few seconds The signal generated by the force due to. standing start The initial response corresponds to a slight. the nylon bolts decayed to zero with a time constant of 19 s. The output signal observed from the force plate therefore. represents the change in the force above that due to the. bolts provided the additional force is applied for a period. significantly less than 19 s, If a force is applied to the center of the plate each piezo. will share the load equally and generate the same charge An. off center force results in an unequal sharing of the load In. commercial force plates the four piezo outputs are monitored. separately and processed electronically to provide measure. ments of the total force as well as the line of action of the. force In the present case all four piezos were connected. directly in parallel Each piezo generates a charge propor. tional to the local force on the piezo When all four piezos. are connected in parallel the output signal is proportional to. the total charge and is hence proportional to the total force Fig 3 The wave form observed when jumping off the floor onto the force. on the plate Consequently the same signal is generated re plate and then stepping off the plate. 305 Am J Phys Vol 67 No 4 April 1999 Rod Cross 305. Fig 4 The wave form observed when stepping onto and then jumping off. the force plate, lowering of the CM in preparation for the jump The im.

pulse is simply related to the height of the jump which was. about 3 cm in this case, IV WALKING WAVE FORMS Fig 6 The wave forms observed when walking a at a slow pace and b a. fast pace The dotted line represents the horizontal component of the ground. Walking or running is not a topic that is usually studied in force in the direction of the gait. a physics course but it is ideally suited for a class of life. science or sports science students It illustrates some inter. esting aspects of elementary mechanics in a way that should ing forward on the ground in order to slow down It is nec. appeal to all physics students For example when walking or essary to slow down by bringing the front foot to rest so. running at constant speed no horizontal force is required at that the back foot can catch up to and then pass in front of. least in a time average sense Why then do we keep pushing the CM While the front foot is at rest the back foot travels. backward on the ground to maintain speed and why do we at about double the speed of the CM The maximum speed of. not keep accelerating with every stride The answer is well a runner is therefore limited to about half the speed at which. known in the biomechanics field but it is probably less well the back foot can be relocated to the front. known by physicists Wind resistance is not sufficient to sup The vertical component of the force acting on one foot. ply the retarding force required to keep the speed constant when walking on the force plate is shown in Fig 6 Also. Top sprinters push backward on the ground with a force shown is the horizontal component in the direction of the. comparable to Mg If wind resistance were the only retarding gait The latter component was not measured with the home. force and if their legs could move fast enough sprinters made force plate but the wave form is well documented in. would reach a terminal velocity comparable to the freefall the biomechanics literature 1 4 7 13 The ratio of the vertical. speed of a person jumping from a plane to the horizontal component indicates that the line of action. The main retarding force in both walking and running of the ground force acts through a point that is close to the. arises from the fact that the front foot pushes forwards on the CM at all times In this way the torque about the CM re. ground resulting in an impulse that is equal and opposite the mains small and the walker or runner can maintain a good. impulse generated when the back foot pushes backwards balance. This is shown schematically in Fig 5 The instantaneous The vertical force wave form is interesting since it has two. walking speed therefore fluctuates but if the average speed distinct peaks where F M g and a dip in the middle where. remains constant then the average horizontal force remains F M g Both peaks are similar in amplitude when walking. zero The retarding force can therefore be attributed to a. at a slow or medium pace but the second peak is smaller. frictional force between the front foot and the ground that. than the first when walking at a fast pace The force rises. prevents the front foot sliding forward along the ground 6. from zero as weight is transferred from the back to the front. This result is surprising since it means that top sprinters as. foot and returns to zero when weight is transferred back to. well as slow walkers must spend about half their time push. the other foot at the end of the stride The force wave form. can be interpreted in terms of the vertical motion of the CM. provided one adds the force on both feet to obtain the result. ant force as shown in Fig 7 It is reasonable to assume that. the left and right wave forms are identical for most people. With the aid of two force plates it is found that the timing of. the ground forces on the left and right feet are such that the. first force peak observed on each foot coincides with the. time at which the back foot lifts off the ground and the. second force peak coincides with the time at which the other. foot first lands on the ground The vertical acceleration a is. Fig 5 A schematic diagram showing the ground forces when running and. obtained by subtracting Mg from the force wave form Inte. the vertical displacement of the CM dashed line Also shown are typical gration of a yields the vertical velocity v and integration of. distances and times when running at 10 ms21 v yields the vertical displacement z The constants of inte. 306 Am J Phys Vol 67 No 4 April 1999 Rod Cross 306. raised to its maximum height as the CM passes over the. stationary foot on the ground At this stage of the walking. cycle the CM describes an arc of radius r 1 m for adults. and the ground force is given by F5M g2M v 2 r Conse. quently if v is greater than about 3 2 ms21 the ground force. drops to zero and the walker becomes airborne or breaks. into a running stride 2 3 For young children r is smaller and. they start running at a lower speed Consequently a child. needs to run to keep up with a fast walking adult, A simplified model of walking is obtained by regarding. the leg on the ground as an inverted pendulum and the leg in. the air as a pendulum suspended at the hip An interesting. demonstration is to hold the top of a meter stick close to the. hip set it oscillating as a pendulum and then walk in step. with the pendulum It is easy to walk faster or slower than. the pendulum but walking in step results in a comfortable. walking speed and presumably helps to minimize the effort. required In general the stride length increases with walking. Fig 7 The vertical component of the ground force F L1R obtained by speed but the cadence i e step frequency increases only. adding the force due to the left foot F L and the force due to the right foot slightly as the walking speed increases 8 9. F R These results were used to calculate the acceleration a ms22 the A sideways oscillation is also observed when walking at a. velocity v ms21 and the vertical displacement z m, frequency that is half the frequency of the vertical oscillation. and slightly out of phase at low walking speeds 4 9 At high. walking speeds the sideways oscillation is in phase with the. gration can be chosen so that the time averages of a and v vertical oscillation Consequently if one observes a walker. are zero and the time average of z is either zero or its mea from behind the CM traces out a shaped Lissajous figure. sured value 7 at low walking speeds or a U shaped figure at high walking. The results of such a calculation are shown in Fig 7 For speeds with a vertical amplitude of about 4 cm and a hori. these calculations the observed force wave forms were ap zontal amplitude of about 4 cm This would also make an. proximated and slightly smoothed by fitting a sixth order interesting and amusing lecture demonstration or video pre. polynomial The total ground force is a maximum when the sentation. CM is at its lowest point and is a minimum when the CM is. at its highest point The amplitude of the vertical displace. ment is typically about 4 cm peak to peak A similar calcu V RUNNING WAVE FORMS. lation based on the horizontal force wave form indicates that Running differs from walking in that both feet are in the. the horizontal speed is a maximum when the CM is at it air for a significant part of the running cycle there is no. highest point and the speed is a minimum when the CM is at period when both feet are on the ground and the feet spend. its lowest point a shorter fraction of the time on the ground as well as a. A surprising result is that z is almost exactly sinusoidal shorter time on the ground each stride In walking the feet. despite the fact that a is distinctly nonsinusoidal and the are off the ground for about 40 of the time so both feet are. force wave form is asymmetric at high walking speeds Since on the ground for a short period each cycle In running each. the a wave form is periodic it can be modeled as a Fourier foot is off the ground for about 70 of the time at a running. series containing a strong fundamental component and speed of 5 ms21 increasing to about 80 of the time at 9. weaker harmonics When this is integrated twice the higher ms21 In running the peak vertical force on each foot while

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