User:Ewold3/sandbox

= Mechanical resonance in biology = Mechanical resonance in biology occurs when a biological system undergoing oscillatory motion experiences a larger response at some frequencies than other frequencies. These special frequencies are called the natural (undamped system) or resonant frequencies (damped system) of the system. Resonance has long been appreciated in the construction of bridges and buildings, which tend to collapse when their natural frequency matches that of oscillations it might routinely encounter due to wind, earthquakes, or human foot traffic. In biology, such resonant mechanics can be beneficial in cycling energy during periodic locomotory movements such as running, hopping, or flapping. This is because a relatively small input oscillation will be amplified and result in a large output oscillation, which is favorable when metabolic energy limited for a particular movement. It can also be detrimental, causing oscillations to be amplified and result in structural collapse of a biological tissue.

Physical Description
For a simple mechanical system consisting of a load with mass m and spring with spring constant k, the natural frequency of the system can be calculated using the following equation. Note that it is inherent to the system, depending only on stiffness and mass.

$$\omega_n=\sqrt{\frac{k}{m}}$$

For a simple mechanical system consisting of a pendulum of length L, the natural frequency of the system can be calculated using the following equation, where g is the gravitational acceleration g=9.81 m/s2. Note that it is independent of initial conditions or mass.

$$\omega_n=\sqrt{\frac{g}{L}}$$

For a simple mechanical system consisting of a load of mass m, spring of spring constant k, and damper with damping coefficient c, the resonant frequency of the system can be calculated using the following equation. Here, $$\zeta$$ is the damping factor. For systems with little damping, the resonant frequency is very close to, although not identical to, the natural frequency.

$$\omega_r=\omega_n\sqrt{1-\zeta^2}$$

$$\zeta=\frac{c}{2\sqrt{km}}$$

In many cases, a system exhibits resonant mechanics at its resonant frequency, but also at whole number multiples of this frequency called harmonics.

Terrestrial Locomotion
Legged locomotion has been described using inverted pendulum models of walking and spring-mass models of running and hopping. Therefore, the equations above can be used to calculate the natural frequencies of a simplified locomotory system on land. During human running and hopping, the effective spring stiffness of the leg can change substantially, suggesting that resonant mechanics may be at play in determining the loading experienced by the leg during locomotion. In isolated muscle-tendon units, matching stimulation frequency to the natural frequency of the passive system resulted in elastic energy savings in in-vitro experiments. However, resonance of soft-tissue components of the human leg may actually dissipate energy upon heelstrike, in a way that is unfavorable for locomotion. Some evidence has been found that muscles in the human leg match the resonant frequency of the soft tissue, serving to dampen this resonant behavior. Brachiating monkeys are modeled similarly using pendula. It has been suggested that they can maintain a close-to-optimal resonant frequency of swinging in accordance with their pendulum dynamics while achieving a variety of different forward speeds.

Swimming
Resonant mechanics have been observed in the aquatic locomotion of jellyfish and jet-propulsive animals like squid. In jellyfish, theoretical and experimental results indicate that speed is maximized when a jellyfish beats its bell at its natural frequency. Further, recent results have shown that turning performance in jellyfish is optimal when the speed of the wave of muscular contraction through the bell is close to the natural wave speed of the bell material. Organisms that locomote by expelling jets of water periodically like scallops have also been found to benefit from resonant energy savings. Bioinspired robotics based on jet propulsive animals have demonstrated the efficacy of resonance in reducing the cost of locomotion underwater.

Flight
The role of resonance in flapping flight remains a topic of active research. Flapping flight at small scales is extremely power-intensive, and in some cases exceeds the power capacity of flight muscles. Weis-Fogh and others in the 20th century postulated that insects may utilize elastic energy exchange via the exoskeleton and wing hinge to overcome this power deficit. The elastic protein resilin has been found in these locations, adding credence to this hypothesis. Experimentally, it has been shown that the wing hinge resonant frequency does not match the wing beat frequency of a variety of insect species, but it has not been explicitly measured whether the thoracic resonant frequency is close to the wing beat frequency of insects.

Plants
Plants are often subjected to periodic forcing due to wind and water currents. Theoretical and experimental results have shown that trees are more susceptible to windsnap when they are driven at their resonant frequencies. However, due to inherent nonlinearities in the material properties of plant tissues, it is thought that the resonant peak for plants is much more spread out than would be predicted from linear theory, suggesting that plants may be able to maintain stability under a wide variety of loading conditions. Kelp and other underwater plants have been shown to experience maximal inertial loading when oscillating at frequencies close to their resonant frequencies.

Hearing
Different components of the human ear use resonance to amplify sonic vibrations. The ability of the ear canal to amplify vibrations of various frequencies depends on the size and geometry of the canal. In adults, the ear canal resonant frequency is between 2000-4000 Hz, although individual components of the ear canal may have higher or lower resonant frequencies. Since resonance changes based on the volume and curvature of the canal, children tend to be more bothered by high frequency sounds due to their smaller ear canals.