A gas spring is a spring mechanism that, rather than using a physical coil to ge

Question

A gas spring is a spring mechanism that, rather than using a physical coil to generate the spring behaviour, uses the properties of a gas to achieve this behaviour. Here is a simplified representation of a gas spring: The force exerted by the gas spring is dictated by the pressure of the gas within it. As the temperature of the gas remains constant, and no gas escapes, the pressure is dictated by Boyle's Law, p_1 V_1 = p_2V_2 = k where p is pressure and V is volume. The force due to the gas is given by F = pA where A is the area of the surface the pressure, and thus the force, applies to (in our case, this area is the cross-sectional area). Note that the air outside the gas spring will also exert a force, but that the pressure for the air will remain constant. As such, the total pressure producing the force is the pressure inside the gas spring minus the air pressure. (*) By considering the forces involved - the force due to pressure in the gas spring and the force due to gravity, as well as air resistance (linear) - show that the motion of mass m is described by (*) Determine the resting length of the spring, x_infinity, when the mass is attached. That is, the length for which all forces balance. (**) Using MATLAB's ode4 5 function, solve the ODE if m = 0.13, c = 0.27, k = 0.35, g = 9.81, and x_0 = 0.1, with x(0) = 0.03 and x'(0) = 0, for 0 lessthanorequalto t lessthanorequalto 5. Plot your solution (**) Write an algorithm for the process of linearising an ODE, then demonstrate your algorithm by applying it to this ODE (around x = x_infinity). (**) Using the same values from part (c), solve the linearised ODE you obtained in part (d) using ode45. Plot this solution on the same axes as your solution to part (c), and comment on the result.

Explanation / Answer

d)   Obtain 2nd ODE by expanding the original ODE as a power series of and then throw away quadratic and other higher order terms in the expansion.

If the eigenvalues are negative or complex with negative real part, then the equilibrium point is a sink . If the eigenvalues are complex, then the solutions will spiral around the equilibrium point.

If the eigenvalues are positive or complex with positive real part, then the equilibrium point is a source . If the eigenvalues are complex, then the solutions will spiral away from the equilibrium point.

If the eigenvalues are real number with different sign (one positive and one negative), then the the equilibrium point is a saddle. For the linear system theses solutions are lines, but for the nonlinear system they are not in general.

e)    Here, x(t) = 0.45 * e-kt   + 0.345 * ekt

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