Section 2 Theoretical considerations

2.1.2 This equation can be visualised as:

(mass x acceleration) + (damping x velocity) + (stiffness x displacement) = dynamic forces

2.1.3 Hence, in order to be able to predict dynamic responses accurately, it is necessary to know precisely mass, damping, stiffness and dynamic forces. As will be described in more detail later in this document, it is difficult to be precise about these parameters. Damping is the most problematical factor (see Ch 1, 5 Damping), particularly since any vibration problems usually occur in way of resonances, where responses are inversely proportional to damping. Vibration excitation forces from engines are normally well defined, those predicted for propellers are less certain. Mass is mostly known accurately, though not absolutely precisely in relation to added mass of sea water that vibrates with a ship, and outfit mass on local deck panels.

2.2.2 Solution of this equation gives roots λ - eigenvalues, and - eigenvectors which are mode shapes. Square roots of eigenvalues λ are natural frequencies in radians per unit time; and hence division by 2π then gives natural frequencies in cycles per unit time.

2.2.3 For each mode, λ = k/m, where stiffness k and mass m are called ‘modal’ or ‘generalised’ values applying to that mode. This represents the value for an equivalent one degree of freedom system, such as a mass suspended on a spring that is constrained to displace in one direction only.

2.2.4 Eigenvectors are normalised usually by selecting each vector element to be divided by the maximum element in the vector, so that the maximum modal displacement is unity.

2.2.5 Eigenvectors are said to be ‘orthogonal’ or ‘normal’ with respect to the mass and stiffness matrices and their use to transform leads to mass and stiffness matrices that are diagonal, which is the so-called modal formulation.

2.3.1 Regarding finite element analysis, in general a significantly more coarse model is adequate for global vibration analysis than is required for stress analysis. Also, in the past, the size of dynamic models needed to be restricted in order to keep computational time and storage requirements within reasonable limits, because solution algorithms available then were such that the number of derived eigenvalues was equal to the number of degrees of freedom in the model. Hence, dynamic models had to be very coarse, or it was necessary to invoke dynamic reduction – in NASTRAN this is described as the Guyan dynamic reduction technique.

2.3.2 Guyan dynamic reduction is basically manual specification of degrees of freedom to be included in an ‘analysis set’ (ASET). The solution process then includes condensation to the analysis set, extraction of eigenvalues, and expansion back to the complete model. Selection of the ASET should be adequate to describe global structural configuration, mass distribution and the vibration modes to be derived. Only translational degrees of freedom need to be included. Provided that the ASET is chosen as previously described, a relatively small set sacrifices very little in accuracy for the global modes.

2.3.3 Later FEA solution algorithms are such that eigenvalue extraction can be restricted to a specified frequency range. However, with increasing use of ship FEA models that were made for stress analysis purposes, which are usually of much finer mesh than is required for global vibration analysis, dynamic reduction is still useful, in order to avoid a plethora of local modes being derived that are not desired or accurately represented.

2.3.4 There are other dynamic reduction methods than the manual ASET selection previously described, such as modal synthesis where substructures are analysed first and then subsequently represented by their vibration modes and combined. However, the degree of control of Guyan dynamic reduction that is afforded to an experienced user is useful. For example, if only main global structure configuration/intersection points of a ship model are specified in the ASET, including enough sections along the length to describe hull vibration modes, then essentially only global vibration modes will be derived. If it were desired to include modes of large deck panels, then points within these can be added to the ASET.

2.4.1 In the direct method, the degrees of freedom are simply the displacements at grid points. The procedure involves the direct solution of the system of equations indicated in Ch 1, 2.1 Basis dynamic equation.

2.4.2 The procedure does not include derivation of vibration modes. It will usually be more efficient for problems in which a large proportion of the vibration modes are required to produce the desired accuracy, which is not generally the case for global ship vibration.

2.4.3 An example of a case where direct dynamic response analysis would be more efficient is the transient response of a LNG containment system to sloshing impacts. In relation to a modal solution, there would be global structure modes, local structure modes, and modes of the containment linings to take into account; thereby many modes over a wide range of frequency would be required in order to obtain sufficient accuracy for the dynamic responses. Hence, a direct dynamic response (transient) analysis would be the preferred choice for such a case.

2.5.1 In the modal method of dynamic problem formulation, the vibration modes of the structure in a selected frequency range are used as the degrees of freedom, thereby reducing the number of degrees of freedom whilst maintaining accuracy within the selected frequency range.

2.5.2 The modal method will usually be more efficient in cases where a relatively small fraction of all of the modes is sufficient to produce the desired accuracy. This is invariably the case for global ship vibration.

2.5.3 The modal method tends to afford more flexible options for specification of damping, which will be described later in this document.

2.6.1 Transient response analysis is dynamic response analysis which is conducted in the time domain, that is, where force versus time is defined. It is most suitable for impact loads or loading of a shock nature, e.g. an explosion.

2.7.1 Frequency response analysis is a dynamic response analysis which is conducted in the frequency domain, that is, where force versus frequency is defined. It is appropriate for loads that are of a continuous sinusoidal ‘steady state’ nature, such as cyclic loads from machinery. Hence, this is the primary method with respect to vibration analysis. It is, of course, possible to define force versus time for cyclic loads from machinery and conduct a transient response analysis, but this is not normally the most efficient method.