Natural Frequency Method: estimating the preferred walking speed of Tyrannosaurus rex based on tail natural frequency.

Pasha A Van Bijlert,Anne S Schulp,A J ‘Knoek’ Van Soest

doi:10.1098/rsos.201441

Abstract

Locomotor energetics are an important determinant of an animal's ecological niche. It is commonly assumed that animals minimize locomotor energy expenditure by selecting gait kinematics tuned to the natural frequencies of relevant body parts. We demonstrate that this allows estimation of the preferred step frequency and walking speed of Tyrannosaurus rex, using an approach we introduce as the Natural Frequency Method. Although the tail of bipedal dinosaurs was actively involved in walking, it was suspended passively by the caudal interspinous ligaments. These allowed for elastic energy storage, thereby reducing the metabolic cost of transport. In order for elastic energy storage to be high, step and natural frequencies would have to be matched. Using a 3D morphological reconstruction and a spring-suspended biomechanical model, we determined the tail natural frequency of T. rex (0.66 s−1, range 0.41–0.84), and the corresponding walking speed (1.28 m s−1, range 0.80–1.64), which we argue to be a good indicator of preferred walking speed (PWS). The walking speeds found here are lower than earlier estimations for large theropods, but agree quite closely with PWS of a diverse group of extant animals. The results are most sensitive to uncertainties regarding ligament moment arms, vertebral kinematics and ligament composition. However, our model formulation and method for estimation of walking speed are unaffected by assumptions regarding muscularity, and therefore offer an independent line of evidence within the field of dinosaur locomotion.

Highlights

Animals display a variety of gaits and walking speeds
The natural frequency of our baseline model of the tail was 0.66 s−1, which we interpreted as the preferred step frequency for T. rex
We have proposed a method based on reconstructing the natural frequency of the vertical swaying of the tail, which we refer to as the Natural Frequency Method

Summary

Introduction

Animals display a variety of gaits and walking speeds. Two locomotor extremes are of particular interest: the maximal speed, and the energetically optimal walking speed, at which the metabolic cost of transport (MCOT, in J m−1 kg−1) is minimal for walking gait. During locomotion at preferred speed, animals tend to make use of resonance by matching locomotor frequencies to the (undamped) natural frequencies of their relevant body parts, sometimes tuned through muscular contractions [6,7,8,9,10,11,12,13]. Based on the commonly adopted idea that PWS is chosen to minimize MCOT [1,2,3,4,5], it has been argued that preferred step frequencies are close to the natural frequencies of relevant body parts [7,11,12]. Determination of the natural frequency of a body part relevant to locomotion can, constrain preferred (and presumably, optimal) step frequencies and walking speeds of extinct taxa [10]

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