Question

This assignment will help you understand the response of a mass-spring-damper or an RLC circuit (or second order ODE) to a step-change in the forcing function. The forcing function for the mass-spring is force, F, and the forcing function for the RLC circuit is voltage, V. In both cases, you will originally assume that the forcing function is zero. At time = 10s, the forcing function is either changed instantneously from 0 to F = 10N due to be addition of mass to the mass-spring-damper system, or it is changed form 0 to V = 10N due to an external voltage (e.g. a battery) to the RLC circuit. You can choose either the mass-spring-damper or the RLC circuit in this assignment.

The gowering equation for the mass-spring-damper system is
m. d²y/dtʸ + μf. dy/dt + ky = Fₑₓₜ (1)
where y is displacement, m is the mass of weight, μf is the friction coefficient, k is spring constant, and Fₑₓₜ is the force generated by the weight
L. d²V/dt² + R. dV/dt + V/C = Vₑₓₜ/C (2)
where V is the voltage drop across the capacitor, L is the indutance, R is the resistance, C is the capacitance, and Vₑₓₜ is the voltage applied to the circuit.

In the case of the RLC circuit, the inductance L is 4H and the capacitance C is 1F. In the case of the mass-spring-damper, the mass is m is 4 kg and the spring constant k is 1 N/m.

Answer 1

The response of a mass-spring-damper system or an RLC circuit to a step-change in forcing function is described by second-order ODEs, with analogous mass-spring and inductance-capacitance components determining the resonant frequency.

Step-by-step explanation:

When a step-change is introduced in the forcing function at a specific moment in time for both a mass-spring-damper system and an RLC circuit, the systems respond with oscillations that are governed by second-order ordinary differential equations (ODEs). For the mass-spring-damper system, the equation is m d²y/dt² + μf dy/dt + ky = F_ext, where y is the displacement, m is the mass, μf is the damping coefficient, k is the spring constant, and F_ext is the external force. For the RLC circuit, the governing equation is L d²V/dt² + R dV/dt + V/C = V_ext/C, where V is the voltage, L is the inductance, R is the resistance, C is the capacitance, and V_ext is the applied voltage. The behavior of these two systems can be analogous, with the mass and spring determining the resonant frequency in the mechanical system, and the inductance and capacitance determining it in the electrical system. The response of each system can be underdamped, critically damped, or overdamped, depending on their respective parameters.