Project - Position control of mass spring damper system

Mass spring damper system:

A spring-mass-damper system is a mechanical system that consists of a mass connected to a spring and a damper. It is commonly used in engineering to model various physical systems, such as the suspension in vehicles, buildings subjected to earthquakes, or vibrating structures.

General equation:

m(d²x/dt²) + c (dx/dt) + kx = F

Spring Damper Transfer function:

Where,

mass is denoted as m, spring constant as k and Damping coefficient as c.

Objective:

To obtain the transfer function of a mass spring damper system and use it in the model. Add a PID controller to adjust the force on mass so that its position follows a reference signal.

Parameters given:

mass, m = 3.6 kg

spring constant, k = 400 N/m

Damping coefficient, c = 100 Ns/m

Thus, natural frequency, wn=sqrt(k/m)

Solution:

The reference signal can be made using a pulse signal generator

Stop Time=27.5 sec

Script file Name: Final_spring.m

Thus, initial gain values are

Kp=K(1)=0

Kd=K(2)=100

Ki=0

Simulink Model:

File name: Final.mdl

Scope result:

As the output signal shows lot of deviation from the refrence signal, we can change the value of Kp to get at value of 1.

(a)Let kp=10000

we can see a lot of change and the signal is becoming stable as compared to Kp=0 output signal.

Still we haven't got required signal output as rising time is more and there is overshoot.

(b) Kp=20000

The overshoot increased but Rise time decreased. Thus, we will increase Kp value further to get 1.

(c)Kp=40000

we got the value of 1, but there is ringing. To minimize ringing we will change Kd value.

(d) Kp=40000, Kd=200

Still there is overshoot.

(e)Kp=40000, Kd=400

Small overshoot is still seen.

(f)Kp=40000, Kd=600

Thus, we got PID controller values as Kp=40000, Kd=600 and Ki=0.

---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

ALTERNATIVE METHOD:

Instead of taking zeta and wn value using given m,k,c values

For underdamped condition we can get PID controller gain values:

zeta=0.5

Script file : Alternative.m

Scope result:

---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Conclusion:

• Obtained the transfer function for the spring mass damper system and succesfully build a PID controlled spring mass damper system model.
• Created a refrence signal as provided.
• Understood how the Kp,Kd,Ki values affect the out signal.
• Was able to adjust the force on mass so that the output signals position follows a reference signal.