Differential Equations in SpringMass Systems
Overview and Examples with
Solutions
What is a Spring-Mass System?
• A spring-mass system consists of a mass
attached to a spring. The system's behavior
can be described by differential equations
based on the presence of damping and
external forces.
1. Undamped Spring-Mass System
• Equation: m d^2x/dt^2 + kx = 0
• - m: Mass of the object
• - k: Spring constant
• Solution: x(t) = A cos(ωt) + B sin(ωt)
• where ω = sqrt(k/m) is the natural frequency.
Example: Undamped Spring-Mass
System
• Given: m = 1 kg, k = 4 N/m
• Equation: d^2x/dt^2 + 4x = 0
• Solution:
• ω = sqrt(4/1) = 2
• General solution: x(t) = A cos(2t) + B sin(2t)
2. Damped Spring-Mass System
• Equation: m d^2x/dt^2 + c dx/dt + kx = 0
• - c: Damping coefficient
• - Types:
• - Underdamped (c^2 < 4mk)
• - Critically damped (c^2 = 4mk)
• - Overdamped (c^2 > 4mk)
Example: Damped Spring-Mass System
• Given: m = 1 kg, c = 2 Ns/m, k = 4 N/m
• Equation: d^2x/dt^2 + 2 dx/dt + 4x = 0
• Damping ratio: ζ = c / (2*sqrt(mk)) = 2/(2*2) =
0.5 (underdamped)
• Solution:
• x(t) = e^(-t)(C1 cos(2t) + C2 sin(2t))
3. Driven (Forced) Spring-Mass System
• Equation: m d^2x/dt^2 + c dx/dt + kx = F(t)
• - F(t): External force applied
• Example: F(t) = F0 cos(ωt)
• Solution involves a particular solution that
matches F(t).