Question

A spring-mass-dashpot system has mass m = 19 kg, damping constant c = 76kg / s and springconstant k = 1900kg / (s ^ 2) Express the ODE for the system in the form

x'' + 2px' + ω₀²x = 0 .
Identify the natural (undamped) frequency of the spring..

Answer 1

The equation of motion for the spring-mass-dashpot system is x'' + 2px' + ω₀²x = 0. The natural frequency of the spring is ω₀ = 10 kg/s.

Step-by-step explanation:

The equation of motion for a spring-mass-dashpot system in the form x'' + 2px' + ω₀²x = 0 is an example of a second-order linear ordinary differential equation (ODE) that describes the behavior of a damped mass-spring system.

In this equation, the values of p and ω₀ can be determined based on the properties of the system. For a system with mass m, damping constant c, and spring constant k, we have p = c / 2m and ω₀ = √(k / m). So, substituting the given values, we have p = 76 kg/s / 2 * 19 kg = 2 kg/s and ω₀ = √(1900 kg/(s^2) / 19 kg) = 10 kg/s.

Therefore, the ODE for the given system is x'' + 4x' + 100x = 0, and the natural (undamped) frequency of the spring is ω₀ = 10 kg/s.