https://simtk-confluence-homeworks.stanford.edu/display/BMH/Homework+2

Due: January 30, 2025

Question 1

(Problem 5.1)

The following equation can be used to calculate muscle fiber force (

Loading

) at an activation (

Loading

), fiber length (

Loading

), and fiber velocity (

Loading

):

Loading

The functions

Loading

and

Loading

represent the normalized active force–length, force–velocity, and passive force–length curves, respectively; and

Loading

represents the maximum isometric muscle force. How does this equation simplify in each of the following cases?

Muscles fibers are:

(a) isometric

(b) at optimal fiber length

(c) maximally activated

(d) passive

Solution (only visible by instructors; please contact us to request access)

Unable to render {include} The included page could not be found.

Question 2

(Problem 6.1)

One way to estimate moment arm of a muscle with respect to a particular degree of freedom is to measure the length of the muscle–tendon complex over a range of joint angles and then use the following equation:

Loading

where

Loading

is the moment arm,

Loading

is the muscle–tendon length, and

Loading

is the joint angle. Prove the above equation. You may prove this with geometry or the principle of virtual work.

Solution (only visible by instructors; please contact us to request access)

Unable to render {include} The included page could not be found.

Question 3

(Problem 6.2)

Moment arms determine the moment-generating potential as well as the amount a muscle changes length with joint rotation. Moment arms vary with joint angle. They are determined by the geometry of the joint: center of rotation, and the relative locations of the attachment points of the muscle with respect to the joint. This question allows you to explore this relationship:

(a) Two important parameters about the moment arm vs. joint angle relationship are: the peak moment arm and the angle at which the moment arm peaks. In order to get a larger peak moment arm, would you want your muscle attachments to be closer to or further from the joint?

(b) Based on the triangle model below, determine the value of the peak moment arm (

Loading

) and the angle (

Loading

) at which the moment arm peaks, in terms of the distances from the attachment sites to the joint center,

Loading

and

Loading

. Assume that

Loading

<

Loading

.

(c) What are the limitations to assuming a model such as this one?

(d) Why are peak moment arm and the angle at which the moment arm peaks important?

Solution (only visible by instructors; please contact us to request access)

Unable to render {include} The included page could not be found.

Question 4

(Problem 6.3)

Muscles Hans and Franz have both entered the prestigious Muscle Olympics (leaving their pesky tendons at home). They will each compete in three events in an attempt to bring home the coveted Golden Sarcomere. Being the poor student that you are, you’re hoping to make a little extra cash by placing some bets on the competition. The Olympic program with the competitor’s vital stats has just arrived in the mail and your bookie is on the phone – what bets do you want to place?

(a) For each event, predict whether Hans or Franz will win. Defend your choice using muscle mechanics principles and equations we’ve discussed in class. (You get credit for your reasons, not for just correctly guessing the winner). If there is a muscle property that you think is important that isn’t listed, assume that it is the same for both Hans and Franz. Assume both muscles have linear force-length properties and moment arms that are constant with joint angle. For all events, the muscles are loaded into a revolute joint with the given moment arm.

	Hans	Franz
Loading (N)	150	150
Loading (cm)	6	6
Moment arm (cm)	2	1.75

“Clean and Jerk”: The joint moves through a range of motion at a constant angular velocity of 100°/sec. The winning muscle is the one that produces the most force at the instant that it reaches optimal fiber length.

“60° Torque-off”: Each joint is fixed at 60°. The winning muscle is the one that produces the largest torque during an isometric contraction. Assume that each muscle is at optimal fiber length when the joint is at 60°.

“110° Torque-off”: Each joint is fixed at 110°. The winning muscle is the one that produces the largest torque during an isometric contraction. Assume again that each muscle is at optimal fiber length when the joint is at 60°.

(b) Hans and Franz were not both satisfied with their performance in the Torque-off competitions—they each thought they should be able to win both the 60° and 110° categories. They convince the Olympic officials to let them have a rematch. Over the weekend, both muscles worked hard to try to improve their odds. Franz went to the local Gold’s Gym and bulked up. Hans went to the local yoga studio and stayed in downward-facing dog all weekend. After this intense conditioning, their vital stats had changed, but each muscle still reaches its optimal fiber length at 60°—get your bookie back on the phone and predict who will win the two Torque-off competitions now.

	Hans	Franz
Loading (N)	150	200
Loading (cm)	9	6
Moment arm (cm)	2	1.75

(c) Using what you learned in this problem, comment on why we may have redundant muscles in our bodies.

Disclaimer: Don’t take Hans’s and Franz’s experiences literally. Only a chronic stretch and a sustained weight training program could actually change the properties of your muscles – but you already knew that.

Solution (only visible by instructors; please contact us to request access)

Unable to render {include} The included page could not be found.

Question 5

(Problem 5.2)

In our dimensionless model of muscle and tendon, we have scaled the cross-sectional area of tendon by peak muscle force. This essentially assumes that the ratio of muscle cross-sectional area to tendon cross-sectional area is constant among various muscles. Find the ratio. Do you think this is a reasonable assumption? State why or why not.

Solution (only visible by instructors; please contact us to request access)

Unable to render {include} The included page could not be found.

Question 6

(Problem 4.1)

Given the neural excitation,

Loading

, shown below, derive an expression for and plot the muscle activation,

Loading

, as a function of time. Label any critical points in your graph. Assume that the time constant for activation is 0.12 seconds and the time constant for deactivation is 0.24 seconds.

Feel free to use either the excitation–activation dynamics ODE from your textbook or the ODE shown below to do your calculations.

Excitation–activation dynamics ODE:

Loading

where

Loading

Solution (only visible by instructors; please contact us to request access)

Unable to render {include} The included page could not be found.

Question 7

(Problem 5.4)

This problem was contributed by Carmichael Ong at Stanford University.

An experiment is performed in which a muscle–tendon unit is fixed at an unknown length, the muscle is activated to a predetermined level, and the tendon force is measured once the muscle and tendon have reached equilibrium (i.e., muscle and tendon forces are equal). The muscle has the following active and passive force–length relationships:

The tendon has the following force–strain relationship:

The muscle and tendon parameters are

Loading

,

Loading

,

Loading

, and pennation

Loading

.

For each of the following experimental conditions (specified activation and measured tendon force), calculate the length of the muscle

Loading

and the length of the tendon

Loading

.

(a)

Loading

,

Loading

(b)

Loading

,

Loading

(c)

Loading

,

Loading

(d)

Loading

,

Loading

Solution (only visible by instructors; please contact us to request access)

Unable to render {include} The included page could not be found.