Question

T+ he mechanical behavior of skeletal muscle tissue can be roughly approximated by the modified Hill model, which is shown in the figure below. CE PS SE F PD ? F X In this model, a contractile force-generating element (CE) is arranged in parallel with a linear spring (PS) and dashpot (PD), which are arranged in series with another linear spring (SE). The contractile element represents the muscle's internal force generation capabilities, and produces nonzero force only when the muscle is activated (i.e. when it receives one or more action potentials). The force-length relationship of this Hill-type model is governed by the following differential equation: $(k_1+k_2)F + \eta \dot{F} + k_2F_m = k_1k_2x + k_2\eta \dot{x}$ Assume that the muscle starts completely from its resting length ($x(0) = 0$). The variables used in this equation are defined as follows: k? = PS spring stiffness ? = PD coefficient k? = SE spring stiffness F = total force produced by the muscle x = total change in length of the muscle Fm = force produced by the CE contractile element 1. Consider the case of an isometric (no change in length) contraction where the contractile element (CE) generates the following force: Fm(t) = F_0(u(t) - u(t-3)) Where F? is the maximum tetanic force (i.e. maximum force produced by the contractile element, and u(t) is a unit step function. a. (3 pts) Define the governing differential equation describing the force-displacement relationship for the passive tissue response. Write your equation in terms of variables F(t), k?,k?, ?, F?, and x(t). b. (5 pts) Classify your differential equation from part 'a' as: i. Ordinary or Partial ii. Homogeneous vs. Non-Homogeneous iii. First-order, second-order, etc. Justify your response. 2. (12 pts) Find an expression for the total muscle force produced by the muscle (F(t)). 3. (5 pts) Considering the different techniques we employed for solving differential equations this semester, identify an alternative technique that could be employed to solve this problem. DO NOT SOLVE. Describe the approach you would take to solve using this alternative technique, and explain why it is better or worse than the approach you used for Part 2.

          T+ he mechanical behavior of skeletal muscle tissue can be roughly approximated by the modified Hill model,
which is shown in the figure below.
CE
PS
SE
F
PD
?
F
X
In this model, a contractile force-generating element (CE) is arranged in parallel with a linear spring (PS) and
dashpot (PD), which are arranged in series with another linear spring (SE). The contractile element represents
the muscle's internal force generation capabilities, and produces nonzero force only when the muscle is activated
(i.e. when it receives one or more action potentials).
The force-length relationship of this Hill-type model is governed by the following differential equation:
$(k_1+k_2)F + \eta \dot{F} + k_2F_m = k_1k_2x + k_2\eta \dot{x}$
Assume that the muscle starts completely from its resting length ($x(0) = 0$). The variables used in this
equation are defined as follows:
k? = PS spring stiffness
? = PD coefficient
k? = SE spring stiffness
F = total force produced by the muscle
x = total change in length of the muscle
Fm = force produced by the CE contractile element
1. Consider the case of an isometric (no change in length) contraction where the contractile element (CE)
generates the following force:
Fm(t) = F_0(u(t) - u(t-3))
Where F? is the maximum tetanic force (i.e. maximum force produced by the contractile element, and u(t)
is a unit step function.
a. (3 pts) Define the governing differential equation describing the force-displacement relationship for
the passive tissue response. Write your equation in terms of variables F(t), k?,k?, ?, F?, and x(t).
b. (5 pts) Classify your differential equation from part 'a' as:
i. Ordinary or Partial
ii. Homogeneous vs. Non-Homogeneous
iii. First-order, second-order, etc.
Justify your response.
2. (12 pts) Find an expression for the total muscle force produced by the muscle (F(t)).
3. (5 pts) Considering the different techniques we employed for solving differential equations this semester,
identify an alternative technique that could be employed to solve this problem. DO NOT SOLVE. Describe
the approach you would take to solve using this alternative technique, and explain why it is better or worse
than the approach you used for Part 2.

T+ he mechanical behavior of skeletal muscle tissue can be roughly approximated by the modified Hill model,
which is shown in the figure below.
CE
PS
SE
F
PD
?
F
X
In this model, a contractile force-generating element (CE) is arranged in parallel with a linear spring (PS) and
dashpot (PD), which are arranged in series with another linear spring (SE). The contractile element represents
the muscle's internal force generation capabilities, and produces nonzero force only when the muscle is activated
(i.e. when it receives one or more action potentials).
The force-length relationship of this Hill-type model is governed by the following differential equation:
(k1+k2)F + ηḞ + k2Fm = k1k2x + k2ηẋ
Assume that the muscle starts completely from its resting length (x(0) = 0). The variables used in this
equation are defined as follows:
k? = PS spring stiffness
? = PD coefficient
k? = SE spring stiffness
F = total force produced by the muscle
x = total change in length of the muscle
Fm = force produced by the CE contractile element
1. Consider the case of an isometric (no change in length) contraction where the contractile element (CE)
generates the following force:
Fm(t) = F0(u(t) - u(t-3))
Where F? is the maximum tetanic force (i.e. maximum force produced by the contractile element, and u(t)
is a unit step function.
a. (3 pts) Define the governing differential equation describing the force-displacement relationship for
the passive tissue response. Write your equation in terms of variables F(t), k?,k?, ?, F?, and x(t).
b. (5 pts) Classify your differential equation from part 'a' as:
i. Ordinary or Partial
ii. Homogeneous vs. Non-Homogeneous
iii. First-order, second-order, etc.
Justify your response.
2. (12 pts) Find an expression for the total muscle force produced by the muscle (F(t)).
3. (5 pts) Considering the different techniques we employed for solving differential equations this semester,
identify an alternative technique that could be employed to solve this problem. DO NOT SOLVE. Describe
the approach you would take to solve using this alternative technique, and explain why it is better or worse
than the approach you used for Part 2.

Added by Jennifer H.

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