ENGD3038 Dynamics and Control

The aim of this activity is to develop an understanding of the dynamic properties of a mass-spring-damper system.

This simulator allows the user to develop understanding of the system physical properties and effect of system parameters on its dynamics.

Description of Assignment

A mass-spring-damper system is simulated; see the front panel of the simulator. You can adjust the force acting in the mass, and the position response is plotted.

The mathematical model of the system can be derived from a force balance (or the Newton's second law: mass times acceleration is equal to the sum of forces) to give the following second order differential equation:

mdx2/dt2 = -Ddx/dt - Kf x + F(t)

1.x [m] is the position of the mass

2.F [N] is the force acting on the mass

3.m [kg] is the mass

4.D [N/(m/s)] is the damping constant (-Ddx/dt is the damping force)

5.Kf [N/m] is the spring constant (-Kfx is the spring force)

It can be shown that the transfer function from force F to position x is a second order transfer function with the following standard parameters:

K = 1/Kf

= D/[2*sqrt(mKf)]

0 = sqrt(Kf/m)

The values of the above parameters can be seen on the front panel of the simulator.

Task

1.Derive the Transfer Function from force F to position x as shown in the Figure in the simulator.

2.The importance of mass m:   With Kf and D constant, vary the mass and write a report on the following.

a.How does the mass m influence the speed of the transient response?

b.How does the mass m influence the damping of the transient response? 

c.How does the mass m influence the stationary response (which is the response as time goes to infinity)?

d.Observe at the front panel how the standard parameters of the second order systems (K, , 0) depend on m. Are the observations in accordance with the above expressions of these parameters?