Two Degrees of Freedom (2DOF) Systems
Explore the behavior of coupled two-mass systems with adjustable parameters.
System Parameters
Mass 1 (Blue)
kg
N/m
Ns/m
m
Mass 2 (Green)
kg
N/m
Ns/m
m
System Information
Natural Frequency 1: 10.00 rad/s
Damping Ratio 1: 0.100
Natural Frequency 2: 10.00 rad/s
Damping Ratio 2: 0.100
Note: The actual system has coupled modes that differ from the individual natural frequencies shown here.
Simulation
Displacement vs. Time
Governing Equations
Equation of motion for mass 1
\[m₁·ẍ₁ + c₁·ẋ₁ + k₁·x₁ + c₂·(ẋ₁ - ẋ₂) + k₂·(x₁ - x₂) = 0\]
Equation of motion for mass 2
\[m₂·ẍ₂ + c₂·(ẋ₂ - ẋ₁) + k₂·(x₂ - x₁) = 0\]
Matrix Form
Where [M], [C], and [K] are the mass, damping, and stiffness matrices
\[[M]{ẍ} + [C]{ẋ} + [K]{x} = {0}\]
Natural Frequencies
Characteristic equation for finding natural frequencies
\[det([K] - ω²[M]) = 0\]