Mass-Spring-Damper
Snapshot from the front panel of the simulator:
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 Newton's second law: mass times acceleration is equal to the
sum of forces) to give the following second order differential equation:
mdx^{2}/dt^{2} = -Ddx/dt -
K_{f} x + F
where
- x [m] is the position of the mass
- F [N] is the force acting on the mass
- m [kg] is the mass
- D [N/(m/s)] is the damping constant (Ddx/dt is the
damping force)
- K_{f} [N/m] is the spring constant (K_{f}x is the
spring force)
It can be shown that the transfer function from force F to position x
is a second order tramsfer function with the following standard
parameters:
K = 1/K_{f}
z = D/[2*sqrt(mK_{f})]
w_{0} = sqrt(K_{f}/m)
The values of the above parameters can be seen on the front panel of the simulator.
The aim of the simulator is to develop an understanding of the dynamic
properties of a mass-spring-damper.
This simulator can develop a
"physical" interpretation of the standard parameters of second order
systems. (This assumes you master the
theory of of second order systems.) However, you may benefit well from
working with most of the tasks below even if you do not master this theory
(you can skip tasks 1d, 2d and 3d).
The mass-spring-damper system is a standard example of a second order
system, since it relatively easy to give a physical interpretation of the
model parameters of the second order system.
Unless otherwise stated, it is assumed that you use the default values
of the parameters.
- The importance of mass m:
- How does the mass m influence the speed of the transient
response
(the transient response is the first part of the response - before
it stabilizes)?
- How does the mass m influence the damping of the transient
response? (There is small damping if the response oscillates
much.)
- How does the mass m influence the stationary response (which
is the response as time goes to infinity)?
- Observe at the front panel how the standard parameters of the
second order systems (K, z, w_{0})
depend on m. Are the observations in accordance with
the above expressions of these parameters?
- The importance of the spring constant K_{f}:
- How does the spring constant K_{f} influence the
speed of the transient response
(the transient response is the first part of the response - before
it stabilizes)?
- How does the spring constant K_{f} influence the
damping of the transient response? (There is small damping if
the response oscillates much.)
- How does the spring constant K_{f} influence the
stationary response (which is the response as time goes to
infinity)?
- Observe on the front panel how the standard parameters of the
second order systems (K, z, w_{0})
depend on K_{f}. Are the observations in accordance with
the above expressions of these parameters?
- The importance of the the damping
constant D:
- How does the damping constant D influence the speed of the transient
response
(the transient response is the first part of the response - before
it stabilizes)?
- How does the damping constant D influence the damping of
the transient response? (There is small damping if the response
oscillates much.)
- How does the damping constant D influence the stationary
response (which is the response as time goes to infinity)?
- Observe on the front panel how the standard parameters of the
second order systems (K, z, w_{0})
depend on D. Are the observations in accordance with
the above expressions of these parameters?
- The importance of the constant force F_{s}: Observe
the step response in x for various F_{s}. How does the
stationary value of the step response, x_{s}, depend on F_{s}?
[SimView] [TechTeach]
Updated 3. June 2009.
Developed by
Finn Haugen.
E-mail: finn@techteach.no. |