11-06-2016, 08:20 PM

Holzer method for natural vibrations. [ HOLZER ]

From the author’s Engineering Collection, included in the ETSII4 module (ETI4 on the CL Library)

This program calculates the natural vibration frequencies of a semi-definite mechanical system with N degrees of freedom using the Holzer method. The vibration can be linear (lineal displacements in the springs) or torsional (angular displacements in the shaft). The vibration modes are also obtained for each natural frequency.

The natural frequencies w are the roots of the frequency function, defined as follows:

g(w) = w^2 SUM { Ij Dj(w) } ; j= 1, 2,… N

Where Dj is also a function of w and of the previous displacements, according to the expression:

Dj = Dj-1 – [ w^2/Kj-1] SUM { Im Dm } ; m= 1, 2,.. j and D1 =1

The terms Kj represent the stiffness constants (elastic or torsional) in the unions between the element masses – typically springs or the shaft depending on the case.

The program offers an initial approximation for the main natural frequency that can be used as guess for the root-finding routine – which is included in the module as well.

Examples.

Calculate the first three natural frequencies and modes of oscillation for a system of 5 rotors connected by a shaft, knowing that the angular momentum is I = 1 kg.m^2 for all of them. The torsional stiffness of the shaft is k = 2 N.m

The solutions are shown on the table below:

From the author’s Engineering Collection, included in the ETSII4 module (ETI4 on the CL Library)

This program calculates the natural vibration frequencies of a semi-definite mechanical system with N degrees of freedom using the Holzer method. The vibration can be linear (lineal displacements in the springs) or torsional (angular displacements in the shaft). The vibration modes are also obtained for each natural frequency.

The natural frequencies w are the roots of the frequency function, defined as follows:

g(w) = w^2 SUM { Ij Dj(w) } ; j= 1, 2,… N

Where Dj is also a function of w and of the previous displacements, according to the expression:

Dj = Dj-1 – [ w^2/Kj-1] SUM { Im Dm } ; m= 1, 2,.. j and D1 =1

The terms Kj represent the stiffness constants (elastic or torsional) in the unions between the element masses – typically springs or the shaft depending on the case.

The program offers an initial approximation for the main natural frequency that can be used as guess for the root-finding routine – which is included in the module as well.

Examples.

Calculate the first three natural frequencies and modes of oscillation for a system of 5 rotors connected by a shaft, knowing that the angular momentum is I = 1 kg.m^2 for all of them. The torsional stiffness of the shaft is k = 2 N.m

The solutions are shown on the table below:

Code:

`w | m1 m2 m3 m4 m5`

---------------------|------------------------------------------

w1 = 0.8740 rad/s; | 1 0.618 0 -0.618 -1

w2 = 2.2882 rad/s; | 1 -1.618 0 1.618 -1

w3 = 2.6900 rad/s; | 1 -2.618 3.2361 -2.61 1