73 Elasticity for Improved Efficiency and Power

Elasticity for Improved Efficiency

The efficiency of the body is improved by it’s ability to store elastic potential energy in the temporary deformation of tissues. For example, when walking and reversing direction you  must slow down, momentarily stop as you change direction, and then speed up again. When you plant your foot against the floor to change direction, your body is able to act kind of like a rubber ball bouncing off a wall and storing some of that original kinetic energy as elastic potential energy in the deformation of tendons and other tissues and then releasing that energy to do some work to get you moving in the other direction. Your muscles then have to use that less chemical potential energy to complete the turn. Your body cannot store and release all of the initial kinetic energy you initially had and some of that mechanical energy will be dissipated (transferred to heat), so this collision is still classified as inelastic. If your body could somehow conserve 100% of the initial kinetic energy then you would bounce back at the same speed you had before the collision, without any muscle input at all, in which case we would say the collision was elastic.

Everyday Example: Reversing Direction

Let’s get a rough estimate of how much energy savings is provided by the elasticity in the body when a 65 kg person walking at 2 m/s reverses direction and walks the other way at 2 m/s.

The change in kinetic energy when stopping is:

    \begin{equation*} \Delta KE = 0 - \frac{1}{2}(65\,\bold{kg})(2\,\bold{m/s})^2 = -130\,\bold{J} \end{equation*}

The tendons of the legs and feet are not able to store all of the kinetic energy as elastic potential energy and some will be turned to thermal energy as friction in joint contacts and jiggling tissue[1]. However, if tendons are able to store and release even 60% of that kinetic energy[2][3], then muscles will only need to provide 40% of that energy to get back up to speed.

The work done by your muscles in that case would be 0.4(130\,\bold{J} = 52\,\bold{J}. At 20% efficiency, the muscles would then need to use an amount of chemical energy:

    \begin{equation*} \Delta PE_c = \frac{52\,\bold{J}}{.2} = 260\,\bold{J} \end{equation*}

Without the store and return elastic energy then muscles would need to supply all 130\,\bold{J} of work to get up to speed. At 20% efficiency, the muscles would then need to use an amount of chemical energy:

    \begin{equation*} \Delta PE_c = \frac{130\,\bold{J}}{.2} = 650\,\bold{J} \end{equation*}

The energy saved by the elastic behavior of tendons was 650\,\bold{J}-260\,\bold{J} = 390\,\bold{J} If this person was a nurse starting and stopping hundreds of times during a shift, that could add up to many tens of thousands of Joules of energy. In the next chapter we will see that walking around for 12 hours will require roughly 12 million Joules of chemical energy, so the energy savings for just the turn arounds are not very significant. However, the same elastic energy storage processes significantly increase the efficiency of simply walking and saves millions of joules of energy during a shift.

Elasticity for Improved Power

In order to examine more closely how energy is stored in the deformation of objects we often model the objects as springs. According to Hooke's Law, the force (F) required to deform a material by a distance (d) from its current shape depends on the distance and the stiffness of the material (k): F = kd. We can model tendons this way, but we may need to use a different value for k when operating in the toe region vs. the linear region of the strain curve because the stiffness of the tendon can be very different in these two regions, as we saw in Unit 5. Springs are designed to follow Hooke’s Law with the same kk through their entire operable range, so the k is often called the spring constant.  For a simple spring with one value of the spring constant, we can calculate the change in elastic potential energy stored in the spring as:

(1)   \begin{equation*} \Delta PE_s = \frac{1}{2}k(d)^2 \end{equation*}

Everyday Example: Jumping Power

In the previous chapter we saw that jump which raises the center of mass of 65 kg person by 0.5 m already requires more power than the muscles alone can typically produce, so how do people jump higher than 0.5 m? Storing elastic potential energy in the strain of the Achilles tendon and releasing that energy to do work at the same time as the muscles can significantly increase the power output. In the Modeling Tissues as Springs chapter of Unit 6 we estimated the typical spring constant of the Achilles tendon to be 8.1 \times 10^6 \,\bold{\frac{N}{m}}. During a jump the tendon may experience strain of more than 0.06, or 6 %. [4]We  can use our equation for elastic potential energy above, but first we need to find the stretch distance corresponding to that strain value for a typical Achilles tendon length of 0.15 m. Starting with the strain equation:

    \begin{equation*} Strain = \frac{\Delta x}{L_0} \end{equation*}

Rearranging for the stretch distance (displacement):

    \begin{equation*} \Delta x = {L_0}\times strain = (0.15\,\bold{m})(0.06) = 9\times 10^{-3}\,\bold{m} \end{equation*}

Ignoring the toe region of the strain curve and treating the tendon as a spring, we can get a rough estimate of the elastic potential energy stored in the tendon during a jump:

    \begin{equation*} PE_E = \frac{1}{2}k(\Delta x)^2 \end{equation*}

Entering values:

    \begin{equation*} \frac{1}{2}\left(8.1 \times 10^6 \,\bold{\frac{N}{m}}\right)(9\times 10^{-3}\,\bold{m})^2 = 324\,\bold{J} \end{equation*}

In the previous chapter we found that a 0.5 m jump required 319 J of energy. Based on this rough estimate, it appears that energy stored in the Achilles tendon could launch a person 0.5 m into the air with no help from muscle power. Timing the release of this elastic energy with both muscle contraction can lead to jumps greater than 0.5 m. Finally, adding pre-stored kinetic energy in the swing of the arms can improve jump height even further (the arms are already moving up so the legs muscles don’t need to provide that kinetic energy during the launch phase). Athletes that focus their training on jumping can achieve greater than 1 m of vertical gain [5]

Elastic Energy Dissipation

Under tension the fibers in your tendons will reconfigure on a microscopic scale, which results in a dissipation of elastic potential energy to thermal energy. Therefore your body cannot store elastic potential energy indefinitely and the elastic boost to efficiency is only available in short bursts. Other animals have adapted to store elastic potential energy for longer periods. Dr. Shelia Patek, Chair of the Biomechanics Division at the Society for Integrative and Comparative Biology, discovered that the mantis shrimp has doubled down on the elastic potential energy strategy by using a “structure in the arm that looks like a saddle or a Pringle chip. When the arm is cocked, this structure is compressed and acts like a spring, storing up even more energy. When the latch is released, the spring expands and provides extra push for the club, helping to accelerate it at up to 10,000 times the acceleration [caused by the] force of gravity on Earth [alone].”[6]

While dissipation, the transfer of mechanical energy to thermal energy, does decrease the efficiency of the body, we will see in the next unit that energy dissipation is critical to preventing injuries.

  1. Sawicki, G. S., Lewis, C. L., & Ferris, D. P. (2009). It pays to have a spring in your step. Exercise and sport sciences reviews, 37(3), 130–138. https://doi.org/10.1097/JES.0b013e31819c2df6 ↵
  2. Tendon elastic strain energy in the human ankle plantar-flexors and its role with increased running speed Adrian Lai, Anthony G. Schache, Yi-Chung Lin, Marcus G. Pandy Journal of Experimental Biology 2014 217: 3159-3168; doi: 10.1242/jeb.100826 ↵
  3. Wiesinger, H. P., Rieder, F., Kösters, A., Müller, E., & Seynnes, O. R. (2017). Sport-Specific Capacity to Use Elastic Energy in the Patellar and Achilles Tendons of Elite Athletes. Frontiers in physiology, 8, 132. https://doi.org/10.3389/fphys.2017.00132 ↵
  4. "Achilles tendon material properties are greater in the jump leg of jumping athletes" by Bayliss, A. J., Weatherholt, A. M., Crandall, T. T., Farmer, D. L., McConnell, J. C., Crossley, K. M., & Warden, S. J., National Library of Medicine, U.S. National Institutes of Heatlh ↵
  5. "Which NBA Player Has the Highest Vertical Jump in NBA History?" by Zack Willis , Sportscasting ↵
  6. "The Mantis Shrimp Has the World’s Fastest Punch" by Ed Yong, Science and Innovation, National Geographic ↵
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