A general theory for the free and forced responses of

The general theory for the free and forced response of strings, shafts, beams, and axially loaded beams is well documented [

Ru [

Kelly and Srinivas [

This paper develops a general theory within which a finite set of parallel structures connected by elastic layers of a Winkler type can be analyzed. The theory shows that the determination of the natural frequencies for uniform parallel structures such as shafts and Euler-Bernoulli beams can be reduced to matrix eigenvalue problems. The general theory is also used to develop a modal analysis for forced response of a set of parallel structures.

The problem considered is that of

Set of

Let

Each structure has the same end supports, and therefore their differential equations are subject to the same boundary conditions. Let

The external forces acing on the

Let

If the stiffness operator is self-adjoint

Clearly each

The standard inner product on

Define

Define

First consider the free response of the structures,

It is well known [_{,}

The mode shape vectors can be normalized by requiring

Since

Let

A special case occurs when the structures are uniform and operators for the individual structural elements are proportional to one another,

The system may be nondimensionalized such that

Assume a solution to (

Equation (

The function

Note that since

Consider the case when

To examine the most general case when

The expansion theorem is used to assume a forced response of the form

A special case occurs when uniform structures are identical such that

The case when one or more of the structures is nonuniform is more difficult in that the differential equations of (

Let

It still may not be possible to solve (

If

Consider

The stiffness operators are of the form of those considered in Section

The eigenvalue problem for the innermost shaft is

First five sets of intramodal natural frequencies of four elastically connected fixed free shafts,

1 | 2 | 3 | 4 | 5 | |

1 | 1.5708 | 4.7124 | 7.8540 | 10.9956 | 14.1372 |

2 | 1.5763 | 4.7142 | 7.8551 | 10.9964 | 14.1378 |

3 | 1.5965 | 4.7210 | 7.8592 | 10.9993 | 14.1400 |

4 | 1.8811 | 4.8247 | 7.9219 | 11.0442 | 14.1750 |

The matrix eigenvalue problem for a set of intramodal frequencies is of the form

As a numerical example, consider four concentric fixed-free shafts connected by layers of torsional stiffness. Solving (

Set of intramodal mode shapes of elastically connected fixed-free torsional shafts with

Set of intramodal mode shapes of elastically connected fixed-free torsional shafts with

Set of intramodal mode shapes of elastically connected fixed-free torsional shafts with

Set of intramodal mode shapes of elastically connected fixed-free torsional shafts with

Suppose that the midspan of the outer shaft is subject to a constant torque,

Forced response of elastically connected torsional shafts at

Forced response of elastically connected torsional shafts at

Now consider the same set of shafts, except that each has a taper, such that the differential equation for the innermost shaft when isolated from the system is

The differential equations for the elastically coupled shafts become

First five sets of frequencies for set of linearly tapered shafts.

1 | 2 | 3 | 4 | 5 | |

1 | 1.639 | 4.736 | 7.868 | 11.006 | 14.145 |

2 | 1.645 | 4.738 | 7.869 | 11.007 | 14.146 |

3 | 1.667 | 4.745 | 7.874 | 11.010 | 14.148 |

4 | 1.981 | 4.861 | 7.944 | 11.010 | 14.187 |

Consider a set of

If the beams are identical (

As a numerical example, consider a set of five fixed-free elastically connected Euler-Bernoulli beams. The solution of Equation (

Using these numerical values, the matrix eigenvalue problem for a set of intramodal frequencies and mode shapes is

The sets of intramodal frequencies are listed in Table

First four sets of intramodal natural frequencies for a set of five elastically connected fixed-fixed Euler-Bernoulli beams.

3.5100 | 22.0300 | 61.7000 | 120.90 | |

1 | 3.4630 | 16.7195 | 44.0798 | 85.7222 |

2 | 6.6167 | 20.6044 | 51.6490 | 99.3829 |

3 | 8.5529 | 23.4906 | 62.2901 | 121.1925 |

4 | 12.7319 | 24.1708 | 62.4633 | 121.3072 |

5 | 14.8317 | 28.0564 | 71.9859 | 139.9667 |

Intramodal mode shapes for a set of five elastically connected Euler-Bernoulli beams.

A general theory is developed for the free and forced response of elastically connected structures. The following has been shown.

The general problem can be formulated using the vector space

If the differential stiffness operator for a single structure is self-adjoint with respect to a standard inner product on

Kinetic and potential energy inner products are defined on both

A normal-mode solution for the free response leads to the formulation of an eigenvalue problem defined for a matrix of operators.

The operator is self-adjoint with respect to the energy inner products leading to the development of an orthogonality condition.

The expansion theorem is used to develop a modal analysis for the forced response.

The case where the structures are uniform and the individual stiffness operators are proportional is a special case in which the determination of natural frequencies and mode shapes can be reduced to eigenvalue problems for matrices on

When the stiffness operators are proportional, the natural frequencies and mode shapes are indexed with two indices, the first representing the spatial mode shape, the second representing the intramodal mode shapes.

If the uniform structures are identical, then a simple formula can be derived for the sets of intramodal natural frequencies using the eigenvalues of the coupling stiffness matrix. The intramodal mode shapes for each spatial mode are the eigenvectors of the coupling stiffness martrix.

An iterative solution must be applied to determine the natural frequencies for the most general case of the uniform structure.

The differential equations for the coupling of identical structures, uniform, or nonuniform can be uncoupled through diagonalization of the coupling stiffness matrix.

Elastically connected uniform strings and elastically connected uniform concentric shafts are applications in which the stiffness operators are proportional.

The differential equations for the concentric shafts, even though they are not identical, can be decoupled when each individual stiffness operator is the same as the individual mass operator.

The individual stiffness operators for uniform Euler-Bernoulli beams are proportional implying that their natural frequencies can be indexed as an infinite number of sets of intramodal frequencies.

The general method is applied here only for undamped systems. However, it can be applied to certain damped systems as well. If the structures are undamped but the Winkler layers have viscous damping, the same