
In the previous lecture we explored the impact of force signal segmentation and showed that care must be taken not to over-simplify (under-segment) the force signal. In fact a safe approach is to utilise all available force data in your simulation.

Care must also be taken when selected a suitable time-step, also known as time discretisation. If too large a time-step is selected, our assumption of a linear change in force magnitude will become less accurate. Again we can demonstrate the influence of varying the time-step by carrying out a sensitivity analysis.

We'll start by generating a function that will generate time and force vectors from a sine wave with varying time-step. This simple functions accepts the desired time-step as an input parameter and returns a time and force vector.

```python
def generateVectors(timeStep):
delta_t = timeStep
nPoints = tmax/delta_t #Number of equally spaced data points
t = np.linspace(0,tmax, int(nPoints)) # Time vector
F_sine = P*np.sin(omega*t) #Force vector
return [t,F_sine]
```

Next we can call the function to generate four different force-time histories with decreasing time-steps.

```python
t1, f1 = generateVectors(0.1)
t2, f2 = generateVectors(0.05)
t3, f3 = generateVectors(0.01)
t4, f4 = generateVectors(0.005)
```

Now, along the same lines as the previous lecture, we'll apply these forces to our SDoF system using our `calculateResponse()` function and observe the response obtained.

```python
Pos1 = calculateResponse(t1,f1, 0.1)
Pos2 = calculateResponse(t2,f2,0.05)
Pos3 = calculateResponse(t3,f3, 0.01)
Pos4 = calculateResponse(t4,f4, 0.005)
```

Plotting the results next...

```python
fig, axes = plt.subplots(figsize=(12,8),nrows=1,ncols=1)
axes.plot(t1,Pos1, label='$\delta_t = 0.1$ s')
axes.plot(t2,Pos2, label='$\delta_t = 0.05$ s')
axes.plot(t3,Pos3, label='$\delta_t = 0.01$ s')
axes.plot(t4,Pos4,'--', label='$\delta_t = 0.005$ s')
axes.set_xlim([0,5])
axes.set_xlabel('Time (sec)')
axes.set_ylabel('Displacement (m)')
axes.set_title('Displacement-time history')
axes.grid()
axes.legend(loc='lower right')
plt.tight_layout()
fig.savefig('timestep.png')
```

<Image
	src="/images/courses/fundamentals-of-structural-dynamics-with-python/time-discretisation/img1.jpg"
	alt="Displacement-time history records obtained using varying time-step size. | EngineeringSkills.com"
	width="1000"
	height="664"
	caption="Fig 1. Displacement-time history records obtained using varying time-step size."
/>

Again we can make some observations:

- We can see that as the time-step is reduced the structural behaviour converges and no further improvement is gained by further time-step reduction.

- Based on this analysis, a sensible time-step would be $\delta t = 0.01$ s. This is because no significant improvement is observed when the time-step is halved, doubling the number of calculations in the simulation.

- In general, with the application of any numerical solution technique, it is critical that the parameter space is explored to confirm the a suitable simulation time-step (time discretisation) and force discretisation is selected.

- For analyses involving spacial parameters (e.g. finite element analysis) or multi-degree of freedom dynamic analysis, space discretisation must also be considered.

Over the last two lectures we've seen that the accuracy of the piecewise-exact method is influenced by the degree of discretisation we use. The main takeaway message from this is that when you are implementing a numerical solution technique, it is always prudent to run multiple simulations in a sensitivity analysis. This will allow you to identify the balance point between sufficient accuracy and computational (and time!) efficiency.

That wraps up our study of general dynamic loading. At this point you should feel comfortable deploying what you've learned in this section and indeed the whole course to your own dynamic problems. In the next and final lecture I'll make a couple of very quick points before signing off.


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