Multiple DOF mass-spring-damper system

Next, we examine the forced oscillation behavior of a mass-spring-damper system with multiple degrees of freedom (M-DOFs).

Simulation Initialization

To initialize the simulation, the following inputs are used:

1. Geometry and connection:
   • The positions of the mass array are, \(\mathbf{x} =[1.0, 2.0, 3.0]^{T}\mathrm{~m}\), and the connection is between the two consecutive mass points. Here, we use \(x_{0} = 0.0\mathrm{~m}\) to build a connection between the first mass point and the initial point.
2. Physical parameters:
   • (i) Mass \({m}_{1} = {m}_{2} = {m}_{3} = 1.0\mathrm{~kg}\).
   • (ii) Damping viscosity \(c_{1} = c_{2} = c_{3} = 0.1\).
   • (iii) Spring stiffness \(k_{1} = 10.0\mathrm{~N/m}\), \(k_{2} = 20.0\mathrm{~N/m}\), \(k_{3} = 30.0\mathrm{~N/m}\).
   • (iv) Natural spring length \(l_{1} = l_{2} = l_{3} = 1.0\mathrm{~m}\).
3. Numerical parameters:
   • (i) Total simulation time \(T=10.0\mathrm{~s}\).
   • (ii) Time step size \(\mathrm{dt}=0.01\mathrm{~s}\).
   • (iii) Numerical force tolerance \(\mathrm{tol}=1\times 10^{-6}\).
4. Boundary conditions:
   • The mass point, \(x_{0}\), is fixed; all other three mass points are free to move.
5. Initial conditions:
   • (i) Initial position \(\mathbf{q}(t=0) = [1.0, 2.0, 3.0]^{T} \mathrm{~m}\).
   • (ii) Initial velocity is set to zeros, \(\mathbf{\dot{q}}(t=0) = [0, 0, 0]^{T} \mathrm{~m/s}\).
6. Loading steps:
   • The periodic force, \(F_{i} \sin (\omega_{i} t) \; \mathrm{with} \; i \in [1,2,3]\), is applied into the system, where the magnitudes of the external force are \({F}_{1}^{\rm ext} = 1.0\mathrm{~N}\), \({F}_{2}^{\rm ext} = 2.0\mathrm{~N}\), \({F}_{3}^{\rm ext} =3.0\mathrm{~N}\) and the frequencies are \(\omega_{1} = 1.0\mathrm{~rad/s}\), \(\omega_{2} = 2.0\mathrm{~rad/s}\), \(\omega_{3} = 3.0\mathrm{~rad/s}\).

Dynamic Rendering