Dynamics: Pendulum Systems

The models demonstrate that the natural frequency of a translational suspended system is independent of its mass and the location of the centre of the mas,s and is the same as that of an equivalent simple pendulum.

Fig. 14-5: A model of a translational suspended system and a rotational suspend system

Fig. 14-5 shows another simple pendulum system. A plate is suspended by four strings through four holes at the corners of the plate. The other ends of the four strings are fixed to two cantilever frames as shown in Fig. 14-5. The strings are vertical when viewed from an angle perpendicular to the plane of the frame and are inclined when viewed in the plane of the frame.

When the plate moves in the plane of the steel frames, it forms a translational pendulum system in which the plate remains horizontal during its motion.

Fig. 14-6: Effects of mass and the centre of mass

Fig. 14-6 shows two similar translational suspended systems in which the masses sway in the plane of the supporting frames. Eight magnetic bars and a steel block are placed on the plate of one suspended system to raise the centre of mass of the system. Applying the same displacement to the two plates in the planes of the frames and releasing them simultaneously, it can be observed that the two suspended systems with different masses and different centres of mass sway at the same frequency. This demonstrates that the natural frequency of a translational pendulum system is independent of its mass and the location of the centre of mass.

The suspended systems in Fig. 14-6 can also be used as two identical rotational suspended systems when the plates sway perpendicularly to the planes of the frames. Applying the same displacements to the two plates in the direction perpendicular to the frames and then releaseing them simultaneously, it can be observed that the plate with the added weights oscilates faster than the other plate.

(a)                                                                                (b)
Fig. 14-7: Effect of the centre of mass

Fig. 14-7 show two arrangements of eight identical magnetic bars. In one case the eight bars stand on the plate (Fig. 14-7a) and in the other case four magnetic bars are placed on the top and bottom surfaces of the plate respectively (Fig. 14-7b). The two cases have the same amount of mass but different centres of mass. Conducting the same experiment as before for the rotational suspended systems, it can be seen that the system with standing bars oscillates faster than that with horizontal bars.

The systems shown in Figs. 14-5 and 14-7b have different amounts of mass but the same location of the centre of mass. If the oscillations of the two systems shown are generated by giving the same initial displacements in the plane perpendicular to the framnes, it can be observed that the two systems oscillate at the same frequency.

The two sets of experiments demonstrate that the natural frequency of a rotational suspended system is dependent on the location of its centre of mass but is independent of the magnitude of its mass.

Table 14-3 compares the times recorded for 30 oscillations of the translational and rotational suspended systems with added masses. Each case was tested twice.

Table 14-3: Comparison of the times for 30 oscillations of the suspended systems
 Translational Suspended Systems (seconds) Rotational Suspended Systems (seconds) Empty (Fig. 14-5) 34.3, 34.4 34.4, 34.5 With full weights (Fig. 14-6) 34.3, 34.5 32.3, 32.4 With magnetic bars placed vertically (Fig. 14-7a) 34.2, 34.2 33.4, 33.3 With magnetic bars placed horizontally (Fig. 14-7b) 34.2, 34.3 34.3, 34.3

Effect of mass on the oscillation of a transverse suspended-system