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The Bode diagram for a mass-spring system  

The Mass-spring system with damping 

For the mass-spring system, the differential equation solver has already been applied to the dynamic equation. Similarly to in electrical engineering, Bode diagrams are widely used in mechatronics.

The Bode diagram program is explained on the control system pages, and it is advised that one study that page first. 

Applied to the mass-spring system, it will give insight into the behavior of the parameters and can be helpful for mechatronic students. To apply the Bode diagram, a transfer function of the differential equation must be derived first.

mass spring system with damping analyzed with the Ti-84 plus CE
differential equation  for maas pring system with damping
transfer function of mass spring system2l.JPG

H(s) = Transfer function

Again we take m=1 kg for the mass, c=6 Nm/s for the damping and k=25 N/m for the stiffness.

In the Ti program we use variable iX instead of variable S where X is the radian frequency (rad/sec)  

transfer function for Ti-84 input

Now we are ready to start up de Bode diagram program on your Ti

Ti-84 input for Bode diagram

The program BodeHOC is written for control system theory, but can also be used to investigate dynamic mechanical systems. After starting, choose 1: new open.The program asks for the transfer function, then the frequency range. The X-axis is logarithmic, so you can choose a wide range; for example, a minimum of 1 and a maximum of 20 rad/sec, since the resonance frequency is around 5 rad/sec.  

After a short time, the Bode diagram appears with amplitude (green) and phase/10 (red). At X=0.6899 the amplitude is -29,36 dB.

On the plots here below, phase and frequency are discussed. The phase at x=0.6999 is -88 degrees at a frequency of 4.897 radians per second, which is calculated with the pink trace (not visible).

With the command "trace" and the "up" and "down" buttons, several curves can be chosen. At a frequency of 4.897 rad/sec, the amplitude is -29.36 dB and the phase is -88 degrees. Finally, the Y-editor created by the Bode program is displayed. 

Bode diagram mass spring system

Bode diagram of a mass-spring system for different damping coefficients 

The Y-editor can be easily adapted to investigate the effect of different damping on the amplitude.

Suppose we want to know the influence on the amplitude for damping c=1, 2, 5 Nm/sec.

By adapting the Y-editor as shown, this can be done easily

Bode diagram for various spring stiffness

The amplitude is strongly influenced by the damping for low damping (c=1Nm/s the amplitude =-13,99dB at a frequency of 5 rad/sec., which is also the theoretical resonance frequency (rad/sec).

Varying the stiffness or the mass can be applied in the same manner. Interested in the Bode program ?

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Since 25-11 2022

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