I am coming from the computer science territory and have not a long trace in mechanics. My background in derivation of the system dynamics could be summarized with utilization of the Lagrange Mechanics but here is a vague point, with which I have confronted, recently. How does the existence of the the jerk term in dynamics of a Mass-Spring-Damper could be justified, where the Lagrange Dynamics just computes up to the 2nd derivatives of the generalized parameters?!…
If one considers a dynamic system, which (from left to right) consists of a spring with constant $k_1$, a mass $m$, a damper with constant $c$ and the other spring with constant $k_2$, all connected together, respectively, the derived dynamics of the system would be declared as: (The junction of spring $k_2$ and damper $c$ is massless)
$$
\frac{cm}{k_2} ยท \frac{d^{3} x}{d t^{3}} +
m \frac{d^{2} x}{d t^{2}} +
c \left( 1 +\frac{k_1}{k_2}\right) \frac{dx}{dt}+
k_1 x= 0
$$
Obviously, the Jerk term has been appeared, up there, noticeably.
Would you please guiding me that how such dynamics could be interpreted by either Lagrange Dynamics or Newtonian Method?!
Best Answer
If I understand your setup correctly, the damper is connected between the mass and the 2nd spring.
Denote the extension of spring 1 with $x_1$ and the compression of spring 2 with $x_2$ (a positive $x_1$ results in a leftward force and likewise for $x_2$).
Since the connection of the damper and spring 2 is massless, the net force there must vanish:
$$0 = -k_2x_2 - c(\dot x_2 - \dot x_1) \Rightarrow c\dot x_2 + k_2x_2 = c\dot x_1$$
At the junction of the mass and spring 1 we have
$$m\ddot x_1 = -k_1x_1 - c(\dot x_1 - \dot x_2) \Rightarrow m\ddot x_1 + c\dot x_1 + k_1x_1 = c\dot x_2$$
Substitute the 1st equation into the 2nd to find
$$m\ddot x_1 + k_1x_1 = -k_2x_2$$
Thus
$$c\dot x_2 = -\frac{cm}{k_2}\dddot x_1 - c\frac{k_1}{k_2}\dot x_1$$
Finally, substitute this into the 2nd equation to yield a third order equation of motion in $x_1$:
$$\frac{cm}{k_2}\dddot x_1 + m\ddot x_1 +c\left(1 + \frac{k_1}{k_2}\right)\dot x_1 + k_1 x_1 = 0$$
A Lagrangian for the system is
$$L = \frac{1}{2}m\dot x^2_1 - \frac{1}{2}k_1x^2_1 - \frac{1}{2}k_2x^2_2$$
The Lagrange equations (of the first kind) are
$$\frac{d}{dt}\frac{\partial L}{\partial \dot x_i} - \frac{\partial L}{\partial x_i} + \frac{\partial D}{\partial \dot x_i} = 0$$
where
$$D = \frac{c}{2}(\dot x_1 - \dot x_2)^2$$
yielding two equations of motion
$$m\ddot x_1 + k_1x_1 + c(\dot x_1 - \dot x_2) = 0$$
$$k_2x_2 - c(\dot x_1 - \dot x_2) = 0$$
which match the first and second equations above.