Response of a damped mass spring system excited by various conditions on...

In this video, we will see how to find the response of a damped mass spring system excited by various conditions.

The governing equation is:
md2x/dt^2+cdx/dt + kx=f

Where m is the mass, c the viscous damping coefficient and k the spring stiffness.

This gives the acceleration
d2x/dt^2=(f- cdx/dt - kx)/m

Using various blocks, we will build our model piece by piece. From a function x, we get dx/dt by differentiating with respect to time. Then we multiply dx/dt by c and x by k and sum the two expressions.

Adding the external force and subtracting cdx/dt+kx then dividing by m we should get the acceleration d2x/dt^2 which is also the result of differentiating dx/dt with respect to time

Working with integrators is better than with differentiator as in the former we could use the initial condition which gives better control on the solution obtained. The model becomes:

On Simulink, we will need the following blocks: two integrators, 3 amplifiers, one constant, one oscilloscope and two blocks to calculate the sum of inputs


The blocks are wired together to give the following model


In fact, the two sum blocks could be merged into one block with 3 inputs and one output. The number of inputs is the number of + and – signs

If we need to recover one of the parameters in Matlab’s workspace, we could send it to the workspace using the block ToWokspace

An object out will be created with two components which are of interest here out.tout is the simulation time and out.x which contains the solution x(t)

Now, let us see what the transient solution is found by Simulink. Obviously, the mass should not be at rest at t=0. Here we choose v0=1.

The solution of the undamped system displayed on the oscilloscope is

If we introduce damping, c=2

Of course, the same plot is obtained if we plot out.x against out.tout

Now let consider a constant force f=10

md2x/dt^2+cdx/dt + kx=10

Notice that 0.1 corresponds to F/k=10/100

Now we will consider a variable force. Let start with a sinusoidal force

We change the constant force in the previous model with a sine source. Also, on the oscilloscope, in addition to the response we display the force

The amplitude is 10 and the excitation frequency is 10 rad/s without phase.

The oscillations at the beginning are the manifestation of the transient behavior. Part of it is due to the initial condition v0=1. After imposing zero initial conditions, we obtain

Most of the transient response has disappeared

Let’s move to a different force, namely the step function. The force appears suddenly at the step time (here 1 s) and remains constant (here 10)

The mass m, initially at rest, remains so until the force appears. Then oscillates around f/k=0.1

Instead of looking for other sources individually or combine some sources to produce special forces, we could use a signal generator which can generate a sine, a square, a sawtooth or random signals.

The response of the system when excited with a square force for instance is

Another interesting source in Simulink is the waveform generator. It enables to define analytically multiple signal expressions and choose one expression as output at a time.

Here we create to signals one of expression sin(10,1,0)+.5*sin(10,2,0) we sum two sinewaves one of amplitude 10 and frequency 1 rad/s and the second of amplitude 5 and frequency 2 rad/s.

The second signal is the previous plus a pulse of amplitude 10 which starts at t=2s and lasts for 3 s.

The chosen output is the first expression

To improve the smoothness of the input force we change the sampling frequency

The result is much better

Now, from our waveform generator, let’s change the out signal to the second expression

We get the response of the system to a combination of two sine waves and one pulse

BTW, to appreciate the usefulness and practicality of the wave generator, to generate the same force we need four sources and a block to do the sum

The first two generate sinewaves of 1 and 2 rad/s. The third source is a step function of amplitude 10 and starts at t=2s and finally the fourth source is step function of amplitude -10 and starts at t=5s (2s+duration of 3s).

Other sources exist in Simulink, but we have seen the main ones. You are advised to start using Simulink and discover all its capabilities.

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