Force-length relationships

Stiffness

Stiffness, in linear cases, is described by Hooke’s Law. Hooke’s law states that the amount an object is stretched or compressed is proportional to the force being applied, and is described by:

𝐹 = 𝑘𝑠 3 where F is the applied force, k is a spring dependent constant that denotes its stiffness, and s is the compression or elongation of the spring element (Young et al. 2008). Graphically, the stiffness of an elastic object can be determined from the slope of the force-

displacement curve (often referred to as a force-length relationship when discussing a biological system, such as muscles). Applying this to the spring mass model of running, the displacement would be the change in length of the spring, defined as the distance from the point of contact with the floor to the CoM, and the force would be the component of the GRF acting in that direction.

Another way of looking at this is that stiffness refers to the rigidity of an object and the extent to which it resists deformation in response to an applied force. Descriptors such as rigidity, flexibility, and compliance are all complementary concepts that are used describe the stiffness of an object. However, each of these terms have strict definitions in

mechanics (Table 3), and their somewhat complacent use can lead to misleading conclusions about how a system works; for example, only the definition of compliance refers to linear behaviour.

Table 3 - Definitions of terms (Parker 2003).

Compliance The displacement of a linear mechanical system under a unit of force

Elasticity The property whereby a solid material changes its shape and size under action of opposing forces, but recovers its original configuration when the forces are removed

The existence of forces which tend to restore to its original position any part of a medium (solid or fluid) which has been displaced

Flexibility The quality or state of being able to be flexed or bent repeatedly Rigidity The quality or state of resisting change in form

Stiffness The ratio of a steady force acting on a deformable elastic medium to the resulting displacement

The shape of a force-length curve can provide information about the type of system being investigated (Figure 28), i.e. spring (linear and mechanical) vs. muscle (non-linear and

biological). A linear force-length curve indicates a ‘mechanically ideal’ system. Therefore, in terms of the spring mass system, the more linear the force-length relationship, the more appropriate the assumption that the body acts similar to a massless linear spring during the stance phase of running. The strength and direction of a relationship can be measured using a correlation coefficient, r, where a value of 1.0 indicates a perfectly linear

relationship. This characteristic will provide one means of testing the first hypothesis, that there will be a linear force-length relationship between GRFs and CoM displacement during the stance phase of running.

Figure 28 - Example force-length relationships. Here the length change is in relative to the resting length of the system, l0, and the relative tension is related to the proof load of the system. The proof load of the system represents the greatest load that can be applied before stretching the system beyond its elastic limit.

Work and energy relationship

Work and energy are interlinked properties. Work is said to be done when the result of a force causing an object to move is a change in the overall energy of the object. It is dependent on the distance that the point of application of the force moves (Young et al. 2008). Physics defines this concept as:

𝑊 = 𝐹 ∙ 𝑠 ∙ 𝑐𝑜𝑠𝜑 4 where φ is the angle between the applied force vector, F, and the direction of

displacement, s. If the displacement has a component in the same direction as the applied force, positive work is done (the energy of the system increases), whereas if the

displacement has a component in the opposite direction of the applied force, negative work is done (the energy of the system decreases) (Young et al. 2008).

Energy refers to the capacity for doing work, and can exist in many forms. In terms of mechanical energy, it is the sum of two components; kinetic energy – that which is associated with a moving an object with mass, and gravitational potential energy – that which is associated with an objects potential to do work due to its location in space (Young et al. 2008). However, when describing springs, the elastic potential energy – that which is associated with the storage of elastic energy due to the deformation of an elastic object – must also be considered.

This elastic potential energy refers to the amount of energy that can be stored in a stretched or compressed spring, and is related back to Hooke’s law and force-length relationships (Young et al. 2008). According to Newton’s third law, if an object applies a force to a spring, an equal and opposite force is applied by the spring on the object, where, k is the spring constant and s is the displacement, i.e. the amount the spring is stretched or compressed.

𝐹𝑜𝑟𝑐𝑒 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑜𝑏𝑗𝑒𝑐𝑡: 𝐹 = 𝑘𝑠 5

𝐹𝑜𝑟𝑐𝑒 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑠𝑝𝑟𝑖𝑛𝑔: 𝐹 = −𝑘𝑠 6

Work done, W, is the product of the force and the distance over which the force is applied. Therefore, since the change in potential energy is the work done in moving an object a distance, the change in elastic potential energy, ΔE, in the spring can be derived as follows (and can be visualised as, or determined from, the area under the force-length curve) (Young et al. 2008):

𝑊 = ∫ 𝑘 ∙ 𝑠 𝑑𝑠 8

𝑊 = 1

2∙ 𝑘 ∙ 𝑠2 9

Δ𝐸 = 𝑊 = 1

2∙ 𝑘 ∙ 𝑠2 10

In perfectly elastic springs, the net elastic potential energy change during loading and unloading would be zero. If, however, a spring is not perfectly elastic, there is an energy loss associated with the deformation, known as the elastic hysteresis (Figure 29). It should be noted that positive work is done (work is being done by gravity) if the force has a component in the same direction as the displacement, i.e. the loading path in Figure 29. While, negative work is done (work is being done against gravity) if the force has a component in the opposite direction to the displacement, i.e. the unloading path in Figure 29. If considering the assumption that the body acts like a single linear spring, then the size of the hysteresis loop determines the amount of elastic potential energy lost, ΔE, during each stance phase of running. Therefore, since the average runner strikes the ground approximately 600 times per kilometre (Lieberman et al. 2010), the total loss in elastic potential energy will presumably contribute to the overall energy cost of running. Thus, the size of the hysteresis loop has the potential to allude to the differences in energy losses between specific groups of runners, e.g. those with different foot strike patterns.

Figure 29 - Example hysteresis loop. Hysteresis determines the amount of elastic potential energy lost during each step, assuming no losses during the aerial phase.