Control System mass spring damper

Sep 27, 2023

Gunter1977

Homework Statement: Hobby project PI controller mass spring damper system

Relevant Equations: mz_ddot + cdz_dot+kz = External force + Force controller

Hello everybody

I have a question about my control system, and I've included a diagram for reference. In the diagram below, I've illustrated three different control schemes using Simulink. The first scheme represents a PI controller for a mass-spring-damper system, which I understand well. It involves calculating the error between the desired value and feedback, and the units appear to be correctly matched.

My question pertains to the control of the PI controller for position or velocity. In the section highlighted in red within the Simulink diagram, I have made an assumption to ensure unit consistency. However, I'm unsure if this approach is permissible. I have experimented with damping coefficients or stiffness constants as conversion factors, but I'm uncertain whether this is the correct approach or if it is allowed

Could someone provide guidance on how to resolve this issue? Have I gone about it the wrong way, and if so, what would be a more appropriate solution?

Your insights and suggestions would be greatly appreciated. Thank you!

Last edited by a moderator: Sep 28, 2023

What is a mass-spring-damper system?

A mass-spring-damper system is a mechanical model that consists of a mass (m), a spring (k), and a damper (c). It is used to describe the dynamics of a system where a mass is attached to a spring and a damper, both of which oppose the motion of the mass. The spring provides a restoring force proportional to the displacement, while the damper provides a force proportional to the velocity of the mass.

How do you derive the differential equation for a mass-spring-damper system?

The differential equation for a mass-spring-damper system is derived using Newton's second law of motion. The sum of forces acting on the mass equals the mass times its acceleration. The forces include the spring force (F_s = -kx), the damping force (F_d = -cv), and any external force (F_ext). The resulting equation is: m * d²x/dt² + c * dx/dt + k * x = F_ext, where x is the displacement, v is the velocity, and a is the acceleration of the mass.

What is the significance of the damping ratio in a mass-spring-damper system?

The damping ratio (ζ) is a dimensionless measure that describes how oscillations in a system decay after a disturbance. It is defined as ζ = c / (2 * √(mk)). The damping ratio determines the behavior of the system: if ζ < 1, the system is underdamped and will oscillate; if ζ = 1, the system is critically damped and will return to equilibrium without oscillating; if ζ > 1, the system is overdamped and will return to equilibrium slowly without oscillating.

How do you determine the natural frequency of a mass-spring-damper system?

The natural frequency (ω_n) of a mass-spring-damper system is the frequency at which the system would oscillate if there were no damping or external forces. It is determined using the formula ω_n = √(k/m), where k is the spring constant and m is the mass. The natural frequency is an important parameter as it characterizes the inherent oscillatory behavior of the system.

What is the role of the damper in a mass-spring-damper system?

The damper in a mass-spring-damper system provides a force that opposes the velocity of the mass, thereby dissipating energy and reducing oscillations. The damping force is proportional to the velocity and is described by the coefficient c. The damper helps control the amplitude of oscillations, ensuring that the system returns to equilibrium more quickly and with less overshoot, which is crucial for stability and