Question 5.17: A Single-Degree-of-Freedom Mass–Spring–Damper System (Energy...
A Single-Degree-of-Freedom Mass–Spring–Damper System (Energy Method)
Reconsider the simple mass–spring–damper system, as shown in Figure 5.29a. Derive the equation of motion by using Lagrange’s equations.
The displacement x is chosen as the generalized coordinate, and the origin is set at the static equilibrium position. Other than the spring force, the system is subjected to the damping force f_b and the external force f (as shown in Figure 5.87), and it is a nonconservative system.
The kinetic energy and the potential energy of the system are T = \frac{1}{2}m\dot{x}^2 and V = \frac{1}{2}k x^2, respectively. The generalized force is
Q = \sum\limits_{j=1}^{2}\mathbf{F}_j \frac{∂\mathbf{r}_j}{∂x} = (f\mathbf{i})\frac{∂}{∂x}(x\mathbf{i}) + (-b\dot{x}\mathbf{i})\frac{∂}{∂x}(x\mathbf{i}) = f – b\dot{x}
Applying the Lagrange’s equation
\frac{d}{dt}\left\lgroup \frac{∂T}{∂\dot{x}}\right\rgroup – \frac{∂T}{∂x} + \frac{∂V}{∂x} = Q
gives the equation of motion of the system
m\ddot{x} + kx = f − b\dot{x}
or
m\ddot{x} + b\dot{x} + kx = f
