Mass-spring-damper model

　・In Japanese
＜Premise knowledge＞
　・Scilab

■Differential equation of spring mass damper system

I will explain how to derive the equation of motion or differential equation of the spring mass damper system, build a model of the spring mass damper, and confirm the operation by simulation. Find the displacement x (t) when the force F (t) is applied to the spring mass damper system as shown in the figure below.



F (t) is the total reaction force from each of the spring, mass, and damper, so the formula is as follows.

The forces applied to each are as follows.

＜Spring equation of motion＞

According to Hooke's law, the force applied to the spring is proportional to the spring constant k and the displacement amount x (t) of the spring, so it is as follows.

＜Damper equation of motion＞

A damper is a part that is attached to a car suspension, for example, and is also called a shock absorber (vibration damping device). The image is like a piston with oil inside a cylindrical part, and even if you push the piston all the way, it does not move immediately, and this plays a role of shock absorption. The image is as follows.


The force applied to the damper is proportional to the speed at which the damper moves, regardless of its position. If you imagine the piston, you can understand this. If you push the piston slowly, the pushing force is small, and if you push the piston quickly, the pushing force is large, which does not change regardless of the position of the piston. Therefore, the force FD (t) applied to the damper can be expressed as follows from the viscosity damping coefficient D and the damper speed v (t).



Also, since the velocity v (t) can be calculated by differentiating the position x (t), the above equation is modified as follows.

＜Mass Equation of motion＞

A mass is just an object with mass, and the relationship between the force "Fm" applied to an object with mass "m" and the acceleration "a" is as follows according to Newton's equation of motion.



Differentiating the position gives the velocity, and differentiating the velocity gives the acceleration, so differentiating the position twice gives the acceleration.

＜Conclusion of differential equations of spring mass damper system＞

By substituting equations (2), (3) and (4) into (1), we were able to derive the differential equation of the spring mass damper.

■Designed a spring mass damper model with Scilab

A little technique is required to design equation (5) with Scilab. Since we want to finally find the displacement amount x (t) when a force is applied, we transform Eq. (5) as follows.



This is expressed in a block diagram as follows.



In this way, dx(t)/dt and x(t) cannot be expressed, but it can be found by integrating dx(t)^2 / dt^2. The following is a block diagram of it.



Expressed in Scilab, it is as follows.

■Simulation result of the operation of the spring mass damper system

The simulation results and each parameter are as follows.

　・k　= 0.5 [N/m] : Spring constant
　・m　= 2 [kg] : Mass
　・D　= 1 [N/(m/s)] : Viscosity damping coefficient

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