The figure below from McMahon compares step length () on a rigid surface (left panel) to step length () on a compliant surface (right panel). The illustration shows the limb in multiple positions during stance phase: (i) at initial ground contact with the knee fully extended; (ii) in mid-stance with the knee flexed; and (iii) at toe-off with the knee again fully extended.

In mid-stance on the rigid surface (left panel), the height of the hip above the ground in mid-stance is , where  is the leg length and  is modeled as a constant, independent of running speed, set by the central nervous system to control the body’s trajectory.

The additional deflection of a compliant surface is characterized by , which represents the peak deflection of the surface in mid-stance. However,  can be approximated as the static deflection that would be observed if the runner were simply standing on the surface.

Figure 3.1 Schematic of a step on a rigid surface (left) and compliant surface (right). Solid line shows the stance leg, broken line shows the swing leg moving forward. Because the foot descends a distance into the compliant track, the step length on the compliant track is necessarily greater. Figure adapted from Figure 5 (McMahon, T. A., & Greene, P. R. (1979). The influence of track compliance on running. Journal of Biomechanics 12(12), 893–904.)

(a) Using the figures and description of  above, derive the expression below for the step length of the runner on a compliant surface, , in terms of the runner’s mass, , runner’s leg length, , the rigid surface step length, , surface stiffness, , and local gravitational acceleration, .

(b) Choose some realistic values for , , , , and . What is the predicted value for ? How does this equation predict how  will vary with decreasing ? Discuss how this prediction compares with your intuition.

We used a spring-mass-dashpot system in the track design problem (i.e., Figure 3.11 in Chapter 3 of the course reader, pictured below).

For that mechanical system, please answer the following questions:

a) Derive an expression for the system’s natural frequency, , with no damping (b=0). Sketch the displacement of the mass vs. time and label the period of oscillation in terms of the natural frequency.

(b) Assuming an underdamped system, derive an expression for the system’s damped natural frequency, , and damping ratio, , when  (rigid track). Sketch the displacement of the mass vs. time, label the period of oscillation in terms of the damped natural frequency. How does the rate of decay of the oscillation relate to the natural frequency, , and damping ratio, ?

(c) The partially completed script simulateTunedTrackSystem.m numerically simulates the 2-degree-of-freedom tuned track system with track compliance and damping. Complete the following sections in the script:

• In the function tunedTrack_equationsOfMotion, use the equations of motion for the spring-mass-dashpot system (see Equation 3.3 in Chapter 3 of course reader) to fill in expressions for  (second time derivative of runner’s displacement, xrdd, in the script) and (first time derivative of track displacement, xtd, in the script).
• In the function simulateTunedTrackSystem, use Figure 3.13 from Chapter 3 of the course reader to assign values to kt_compliant, kt_tuned, and kt_stiff, which represent the stiffnesses of compliant, tuned, and stiff tracks, respectively.

(d) Run the completed script to generate plots of the runner’s and track’s displacement over time. Assuming foot contact time is half the period of oscillation of the runner, use the graph to estimate foot contact time () for the three stiffness you chose. Compute normalized foot contact time () as contact time divided by the rigid track contact time. Using the same set of three stiffnesses, run the script for two different masses. What is the effect of the runner’s mass on contact time and normalized contact time?

% In MATLAB, the main function (in this case, simulateTunedTrackSystem)
% must match the file name exactly. To that end, do not rename the file!
% Do not modify the file except in as noted in the two steps below:
% Step 1: Fill in the two lines where noted in tunedTrack_equationsOfMotion
% Step 2: Fill in the three lines where noted in simulateTunedTrackSystem

function simulateTunedTrackSystem
    % ODE parameters (end time, initial conditions)
    t_final = 5;
    xr_initial = 0;
    xrd_initial = 0.1;
    xt_initial = 0;

    % System parameters
    m = 50;
    km = 1000;
    zeta = 0.55;

    b = 2*sqrt(m*km)*zeta;

    %-- FILL IN THE THREE LINES BELOW, e.g., kt_compliant = 0.1*km; --%
    kt_compliant = %-- FILL IN HERE --%
    kt_tuned = %-- FILL IN HERE --%
    kt_stiff = %-- FILL IN HERE --%

    % Use Matlab's ode45 to integrate equations of motion
    % Syntax:
    %   [t_solution, X_solution] = ode45( <function handle to equations of motion>, [t_initial, t_final],

initial_conditions)
    [time_compliant, X_compliant] = ode45(@(t,X) tunedTrack_equationsOfMotion(t, X, m, km, b, kt_compliant),

[0, t_final], [xr_initial, xrd_initial, xt_initial]);
    [time_tuned, X_tuned] = ode45(@(t,X) tunedTrack_equationsOfMotion(t, X, m, km, b, kt_tuned), [0, t_final],

[xr_initial, xrd_initial, xt_initial]);
    [time_stiff, X_stiff] = ode45(@(t,X) tunedTrack_equationsOfMotion(t, X, m, km, b, kt_stiff), [0, t_final],

[xr_initial, xrd_initial, xt_initial]);

    % Plot results
    figure
    subplot(2,1,1)
    plot(time_compliant, X_compliant(:,1)), hold all
    plot(time_tuned, X_tuned(:,1))

    plot(time_stiff, X_stiff(:,1))
    plot([0,t_final], [0,0], 'k--')
    grid on
    ylabel('x_r')
    title('Displacement of runner')
    legend('Compliant track', 'Tuned track', 'Stiff track')

    subplot(2,1,2)
    plot(time_compliant, X_compliant(:,3)), hold all
    plot(time_tuned, X_tuned(:,3))
    plot(time_stiff, X_stiff(:,3))
    plot([0,t_final], [0,0], 'k--')
    grid on
    ylabel('x_t')
    title('Displacement of track')
    xlabel('t')

function Xd = tunedTrack_equationsOfMotion(t, X, m, km, b, kt)
    % This function implements the equations of motion for the 2-degree of
    % freedom tuned track system.
    %
    % Variables:
    % xr: runner displacement
    % xrd, xrdd: first and second time derivatives, respectively, of xr
    %    i.e., xrd = d(xr)/dt, and xrdd = d^2(xr)/dt^2
    % xt: track displacement
    % xtd: first time derivative of xt
    %    i.e., xtd = d(xt)/dt

    % System state (X), and state time derivative (Xd) % X = [xr, d(xr)/dt, xt] 
    % X = [xr, d(xr)/dt,xt]
    % Xd = [d(xr)/dt, d^2(xr)/dt^2, d(xt)/dt]
    %
    % The equations of motion from dynamics are:
    %    m(xrdd) = km(xt - xr) + b(xtd - xrd)
    %    km(xt - xr) + b(xtd - xrd) = -kt(xt)

    % Get state values from X
    xr  = X(1);
    xrd = X(2);
    xt  = X(3);
   
   % Compute state derivatives
   %-- FILL IN THE TWO LINES BELOW --%
   %-- Remember to use the same variables names from above! --%
   xtd = %-- FILL IN HERE --%
   xrdd = %-- FILL IN HERE --%

% Return vector of state derivatives
    Xd(1,1) = xrd;
    Xd(2,1) = xrdd;
    Xd(3,1) = xtd;
end

The Froude number (sounds like “food” with an “r”) is a useful parameter to study how locomotion changes with body size. It is a dimensionless variable that relates a scalar dimension of the body (typically leg length) to the forward velocity of the body. Theoretically, animals of different sizes that rely on pendulum-like mechanics for gait will use the same form of locomotion at a given Froude number (e.g., a T-Rex and a chicken traveling at the same Froude number would look the same). The Froude number is defined as:

where  = the speed of movement (in m/s)

= the acceleration due to gravity (in m/s)

= the effective length of the limb (in m)

(a) Using the simple inverted pendulum model from the previous question, explain why the walk-run transition theoretically occurs at . (Note: the model is redrawn below with the linear speed  and gravitational acceleration  shown.)

(b) Dr. Evil (a somewhat normal adult male with a leg length of 0.9m) and Mini-Me (an exact 1/8 replica of Dr. Evil) need to go to the other side of their secret lair in Nevada to feed their sharks and repair some laser beams. Dr. Evil demands that Mini-Me hold his hand so they can remain side-by-side as they travel to the shark tank. Dr. Evil walks to the shark tank at optimum walking speed (Fr = 0.25). Once arriving at the destination, Mini-Me says that he is too tired from running to keep up with Dr. Evil to feed the sharks. How can you explain this?

(c) Dr. Evil is tired of having a miniature version of himself and orders the biomechanical engineers on his lunar base to build a robot that looks, walks, and behaves exactly how he does on Earth. Given that gravity = (1/6)gravity, why is the lunar engineering team doomed to failure? 

(d) Walking speed is a good indicator of mobility and is often used in research of gait disorders. It has been suggested that walking speed should be reported in terms of the Froude number to remove the effects of individual subject height in experiments. Suppose a patient with cerebral palsy walks with excessive knee flexion (this is called crouch gait). Knowing that she needs leg length to compute Fr, the clinician measures the patient's leg length while he lies on the exam table. She also measures his walking speed. She is pleased when she computes Fr = 0.25, which suggests a reasonable walking speed for the patient’s stature and thus good energy efficiency. However, after measuring the patient's oxygen consumption, she finds that his gait is not energetically efficient at all. Explain this discrepancy in her results using the Froude number. Based on this information, do you think that Fr is a good clinical indicator of walking speed?