Mechanics

Homework

1-) A linear muscle mechanics is shown in figure below. Here, the elastic element, Cp, is placed in parallel to the viscous damping element R and the contractile element, and the entire parallel combination is placed in serieswith the elastic element, Cs, and the lumped representation of the muscle mass, m. Derive an expression for the transfer function relating the extension of the muscle, x, to an appliedforce, F. Convert this transfer function description into the equivalent statespace model.

2-) Figure below displays the equivalent circuit of a short length of squid axon according to the Hodgkin-Huxley model of neuronal electrical activity. The elements shown as circles represent voltage sources that correspond to the Nemst potentials for sodium, potassium, and chloride ions. The resistances are inversely proportional to the corresponding membrane conductances for these three types of ions, while C represents membrane capacitance. Derive the Hodgkin-Huxley equation, i.e., the differential equation that relates the net current flowing through the membrane, I, to the applied voltage across the membrane, V.

3-) Figure below shows a physiological model of system. For this system, drive differantial equation of the system. Show the block diagram of the system. Determine the transfer function. Determine the impulse response. Determine Vo(t) fot t>0; if input voltage Vi=e-3tu(t).

4-) Figure below shows the block diagram of a sophisticated biomedical device for regulating the dosage of anesthetic gases being delivered to a patient during surgery. Note that the plant and controller are themselves feedback control systems.

(a) Derive an expression for the open-loop gain of the overall control system.

(b) Derive an expression for the closed-loop gain of the overall control system.

(c) If GI = 1, G2 =2, HI = I, and H2 =2, what is the loop-gain of the overall system?