Let’s have a little fun and experiment with moving your camera. In essence we mostly intend to get a clear sharp, well composed and focused image. Going out and on intent messing up your shot as you move your camera or lens against all rules. There are a number of ways to get different and quite pleasing effects with different techniques.
The main thing you need to do is setting your shutter speed to a longer exposure than you usual would do. That's the basis for starting, but getting some good effect takes practice, practice and lots of experimentation and on top luck, luck, luck.
Keeping the Camera still at a low shutter speed and zooming your lens at the same time will give you the zooming effect like this one.
As you want to capture only the zooming movement any side to side or up and down shake will impact the smoothness of the lines. If there is too much movement the shot will look too blurry. To eliminate this unwanted shakes use a tripod. Low Light will help as you are using longer shutter speeds which let more light into your camera.
What shutter speed should you use? There is unfortunately no specific shutter speed that will work for all situations. Factors include levels of light, speed of zooming the lens etc. I mostly shoot at up to a second (or even a little longer) which is usually enough to zoom a lens from one end to another. Just experiment with different shutter speeds to see what works best.
Flash – and using it in adding light directed to a certain part of the image helps tremendous. You can do this with different light sources but the most common one is flash which should be directed just to some parts of the image. Restrain the flash light with flags to direct it for example to a face. You get what you see. Fire it during your long exposure a rear curtain mode and you can freeze part of the moving image.
Apart from zooming with your lens you can turn or I might say spin your whole camera to get the following effect. Working with an assistant who points the flash at the face got me this nice image.
Another technique which I usually apply on flowers or trees in a forest can work as well on other shots as you see but this is a rare lucky exception. The former named subjects are much more forgiving and the image is more pleasing. It works quite well in architecture shots at night I might say.
So get out, experiment and hopefully get some good results. I got about five pleasing images in my humble opinion out of one hundred taken. So as said in the beginning a lot of experimenting and luck.