Natural Frequencies Trend of Continuous System

Mode of Vibration

A mode of vibration can be defined as a way of vibrating, or a pattern of vibration, when applied to a system or structure that has several points with different amplitudes of deflection.

From: Encyclopedia of Vibration , 2001

MODE OF VIBRATION

D.J. Ewins , in Encyclopedia of Vibration, 2001

Definition of Mode of Vibration

A mode of vibration can be defined as a way of vibrating, or a pattern of vibration, when applied to a system or structure that has several points with different amplitudes of deflection. A mode of vibration comprises two distinct elements: first, a time variation of the vibration and, second, a spatial variation of the amplitude of the motion across the structure. The time variation defines the frequency of oscillations together with any associated rate of decay or growth. The spatial variation defines the different vibration amplitudes from one point on the structure to the next. The underlying expression that defines a vibration mode can be written as:

(1) $\begin{array}{c}\hfill x(t)={\mathrm{Xe}}^{\mathrm{st}}\\ \mathrm{or}:\hfill \\ \hfill x(y,t)=X(y){\mathrm{e}}^{\mathrm{st}}\end{array}$

depending upon whether the system is discrete, described by a finite number of specific degrees of freedom (Figure 1A), or continuous, in which case the motion is defined by a continuous function of position, as illustrated in Figure 1B. In these expressions, the coefficient s represents the time-dependent property while the vector X or the function X(y) represents the spatial dependence of the vibration mode. Throughout this entry, it will be convenient to focus discussion on the former of these two expressions, the discrete version, as this is the one in more widespread use. However, it should be noted that all properties are common to both discrete and continuous versions.

Figure 1. (A) Discrete and (B) continuous systems.

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Propeller Blade Vibration

J.S. Carlton FREng , in Marine Propellers and Propulsion (Fourth Edition), 2019

Abstract

The modes of vibration of a propeller blade, beyond the fundamental and first torsional and flexural modes, are extremely complex. This complexity arises from the nonsymmetrical outline of the blade, the variable thickness distribution, both chordally and radially, and the twist of the blade caused by changes in the radial distribution of pitch angle. In addition, the effect of the water in which the propeller is immersed causes both a reduction in modal frequency and a modified mode shape when compared with the corresponding characteristics in air. To introduce the problem of blade vibration, it is easiest to consider the vibration of a symmetrical flat blade form in air because, in this way, many of the practical complexities are eliminated for a first consideration of the problem.

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Vibrations Induced by Internal Fluid Flow

In Flow-induced Vibrations (Second Edition), 2014

4.2.1.4.2 Fluid added mass and natural frequency of transverse vibration of single bellows

Transverse vibration modes of single bellows are shown in Fig. 4.22. The equation to calculate the mechanical natural frequency of the bellows transverse vibration [29] based on the Timoshenko beam model is expressed considering bellows rotary inertia and fluid added mass due to bellows distortion. The total mass per unit length m _tot of the bellows is given by the following equation:

(4.21) $m_{\mathrm{tot}}=(m_{\mathrm{f}}+m_{\mathrm{b}})$

(4.22) $m_{\mathrm{b}}=\frac{4\pi R_{\mathrm{m}}}{p}(h+0.285p)t{\rho }_{\mathrm{b}}$

(4.23) $\begin{array}{ll}\hfill m_{\mathrm{f}}& =(m_{\mathrm{f}1}+m_{\mathrm{f}2})\hfill \\ \hfill & =\left\{\pi {\left(R_{\mathrm{m}}-{\displaystyle \frac{h}{2}}+{\displaystyle \frac{2h{R}_2}{p}}\right)}^2+{\alpha }_{\mathrm{f2k}}\mu R_{\mathrm{m}}^3\right\}{\rho }_f\hfill \end{array}$

$m_{\mathrm{f}1}=\pi {\left(R_{\mathrm{m}}-\frac{h}{2}+\frac{2h{R}_2}{p}\right)}^2{\rho }_{\mathrm{f}}$

$m_{\mathrm{f}2}={\alpha }_{\mathrm{f}2\mathrm{k}}\mu R_{\mathrm{m}}^3{\rho }_{\mathrm{f}}$

${\alpha }_{\mathrm{f}2\mathrm{k}}=0.066\frac{A_{1\mathrm{k}}^2}{l^4}{\left(R_{\mathrm{m}}-\frac{h}{2}\right)}^{2}p$

where m _b is the bellows mass per unit length and m _f the fluid added mass per unit length. The fluid added mass m _f is composed of the mass of fluid per unit length of the bellows m _f1 and the convolution distortion component m _f2. The added mass coefficient μ is presented in Fig. 4.24.

Figure 4.24. Half-convolution added mass coefficient for transverse vibration [29,31].

The equation of motion for bellows transverse vibrations is:

(4.24) $E{I}_{\mathrm{eq}}\frac{{\partial }^{4}w}{\partial x^4}+m_{\mathrm{tot}}\frac{{\partial }^{2}w}{\partial t^2}-\rho I\frac{{\partial }^{4}w}{\partial x^2\partial t^2}+P\pi R_{\mathrm{m}}^2\frac{{\partial }^{2}w}{\partial x^2}=0$

where EI _eq is the equivalent bending stiffness, w the transverse displacement, and P the fluid pressure.

Considering the boundary conditions of the bellows, the k-th transverse mechanical natural frequency of the bellows, f _k, is given by the following equation:

(4.25) $f_{\mathrm{k}}=\frac{R_{\mathrm{m}}}{4\pi l^2}{A}_{1\mathrm{k}}{\left[\frac{k\mathit{p}-4\pi l^{2}P{A}_{2\mathrm{k}}}{m_{\mathrm{tot}}+{\displaystyle \frac{{\rho }_{\mathrm{l}}I}{l^2}}{A}_{\mathrm{4k}}}\right]}^{1/2}$

$\rho I=\pi R_m^3\left(\left(\frac{2h}{p}+0.571\right)t{\rho }_b+\frac{h}{p}(2R_2-t){\rho }_f\right)$

The coefficients A _1k, A _2k, A _4k for the first four modes of single bellows are given in Table 4.3.

Table 4.3. A _ik constants for transverse modes [29]

Mode no. k	1	2	3	4
A 1k	22.37	61.67	120.9	199.9
A 2k	0.02458	0.01211	0.00677	0.00374
A 4k	12.30	46.05	98.91	149.4

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Sound Radiation from Wheels and Track

David Thompson , in Railway Noise and Vibration, 2009

6.3.1 Radiation ratio

The modes of vibration of a railway wheel have been described in Chapter 4. For a typical wheel radius of 0.46   m, the condition ka  =   1 corresponds to a frequency of 120   Hz. It can therefore be expected that its radiation ratio will be close to unity for most frequencies of interest. Moreover, as it is relatively thick, the structural wavelengths are not particularly short and acoustic short-circuiting effects only occur at relatively low frequencies.

In [6.7] Remington simply used σ  =   1 for the wheel radiation ratio (it was actually stated in the form σ  =   2, as the surface area S in the equation was taken only as the area of one side of the wheel). Earlier, in [6.8], he also used an expression for the radiation ratio of a rigid unbaffled disc, which gave a lower radiation at low frequency particularly below about 100   Hz.

This approach does not take account of the vibration distribution due to the mode shape. Other early models of the sound radiation from railway wheels [6.9, 6.10] made use of the Rayleigh integral technique [6.2] but this does not give the correct radiation ratio at low frequencies. For example, for a simple n  =   0 mode (i.e. with 0 nodal diameters) the Rayleigh integral would predict the radiation ratio as that of a monopole, where the true result is that of a dipole, as discussed below.

The boundary element method (BEM) [6.11] can be used in order to take account of complex geometry. This is a numerical technique in which the vibrating surface is represented by a mesh of elements and the sound field is calculated by solving the Helmholtz integral equation in discrete form. Unlike the acoustic finite element method, it readily allows for infinite domains. Fingberg [6.12] used the boundary element method to predict the sound radiation from a wheel in its various normal modes. Making use of the axisymmetry of the wheel, the wheel was represented only by its cross-section. It was shown that the radiation ratio can be significantly smaller at low frequencies than that predicted by the Rayleigh integral method. At high frequencies, where the radiation ratio is close to unity, variations of around 2   dB were found. Experimental validation was performed on a model wheel at scale 1:5.

A similar BEM approach was used in [6.13] but the radiation ratio due to vibration in each normal mode was calculated for a range of frequencies rather than just at the corresponding natural frequency. These results will be summarized here.

The modes of vibration of a wheel were calculated first using the finite element method, see Chapter 4. The radiation ratios obtained for the zero-nodal-circle axial mode shapes with different numbers of nodal diameters, n (see Figure 4.3) are shown in Figure 6.6. In each case, the radiation ratio rises sharply at low frequency, reaching unity between about 250 and 1250   Hz, and then oscillating at higher frequencies.

FIGURE 6-6. Radiation ratio of zero-nodal-circle axial modes of 920   mm freight wheel with various numbers of nodal diameters, n. Thick lines: —, n  =   0; – – –, n  =   1; ····, n  =   2; – · –− ·, n  =   3; thin lines: —, n  =   4; – – –, n  =   5; ····, n  =   6, – · – ·, n  =   7

The result for the n  =   0 mode has a frequency dependence at low frequency of f⁴, reaching unity at about 250   Hz. For other modes of the wheel, the slope of the low frequency part of the curve increases with increasing n. For n  =   1 it increases at a rate of f⁶, for n  =   2 at f⁸, etc. Thus in general it follows f²⁽ⁿ⁺²⁾.

In order to explain these slopes, Figure 6.7 shows a number of wheel modes schematically. The upper row shows a disc set in an infinite baffle. The n  =   0 mode is a simple volume source which acts as a monopole; the n  =   1 mode has regions either side of the node line that are vibrating out of phase with each other, thus forming a dipole at low frequencies, where the wavelength is long compared with the disc dimensions. Similarly, the n  =   2 mode has two positive and two negative source regions which together form a quadrupole at low frequencies.

FIGURE 6-7. Indication of the equivalent multipoles for various wheel modes. Top: modes of a disc set in a rigid baffle; middle: modes of an unbaffled disc; bottom: radial modes

FIGURE 6-8. Radiation ratio of one-nodal-circle axial modes of 920   mm freight wheel with various numbers of nodal diameters, n. Thick lines: —, n  =   0; – – –, n  =   1; ····, n  =   2; – · – ·, n  =   3; thin lines: —, n  =   4; – – –, n  =   5; ····, n  =   6

In the second row of Figure 6.7 axial modes are shown of a wheel in free field. Here, in addition to the positive and negative source regions seen for the baffled disc, the front and rear of the wheel correspond to source regions of opposite polarity. Thus the n  =   0 mode now forms a dipole, the n  =   1 mode a quadrupole, etc. The n  =   0 mode has similar behaviour to the oscillating sphere, see Figure 6.3, but the result is lower; the correct behaviour can be found by using a reduced radius of 0.3   m in equation (6.7). A similar phenomenon has already been noted for an oscillating disc.

The frequency dependence of the radiation ratio for these various modes can thus be explained in terms of the characteristic frequency dependence of simple sources. As the mode order, n, increases, the order of the multipole increases. The frequency at which σ becomes equal to unity also increases as n is increased. This can be related to the size of the component simple sources on the wheel surface, which reduce as the wavelength around the wheel reduces.

It may be noted that the wavelengths in the wheel are such that the critical frequency is very low and no acoustic short-circuiting occurs. The wheel web has a typical thickness of 25   mm and a flat steel plate of this thickness would have a critical frequency of 500   Hz (equation (6.16)). The zero-nodal-circle axial modes have wavelengths that are determined by the wheel tyre which has a thickness of 135   mm and is even stiffer than the web.

Figure 6.8 shows corresponding BEM results calculated for one-nodal-circle axial modes. In these mode shapes the maximum axial motion is in the web region; the tyre has very little axial motion (see Figure 4.3). These results are very similar to those for the zero-nodal-circle axial modes, although for frequencies below about 500   Hz they are approximately a factor of 2 lower. Apart from this factor, the number of nodal diameters, n, is more important in determining the radiation ratio than the deformed shape of the cross-section.

Due to the curved shape of the wheel web, the predominantly radial modes of this wheel are found to contain considerable axial motion, similar in form to the one-nodal-circle modes, see Figure 4.3. Their radiation ratios are consequently rather complex. The radiation ratio is therefore presented for a purely radial motion of the tyre, in which the web is assumed not to vibrate [6.13]. This result is shown in Figure 6.9. Here the radiation ratio does not reach unity until close to 1   kHz. At very low frequencies the slope of each curve is less than those in Figure 6.6. For the n  =   0 motion the low frequency slope is found to correspond to f². This motion is a radial in-phase pulsation of the whole tyre, which can be seen to be equivalent to a monopole, see Figure 6.7. For n  =   1 motion, the tyre oscillates vertically, which thus corresponds to a dipole and the slope of the curve is f⁴. For each value of n, the order of the multipole approximation at low frequency is one less than for the corresponding axial motion and the slope is f²⁽ⁿ⁺¹⁾. However, there is also a region where the curves appear to drop below this trend. This can be associated with partial cancellation between the inner and outer faces of the tyre.

FIGURE 6-9. Radiation ratio for radial motion of the tyre of 920   mm freight wheel with various numbers of nodal diameters, n. Thick lines: —, n  =   0; – – –, n  =   1; ····, n  =   2; – · – ·, n  =   3; thin line: —, n  =   4

Although these results have been given for each mode shape for a wide range of frequencies, allowing the radiation due to forced vibration to be determined from them, it is worth noting that most modes of vibration of the wheel occur in the region where the radiation ratio is close to unity. Figure 6.10 shows the results calculated using BEM for the (measured) natural frequencies of each mode. These are all within 1   dB of unity except for the low order zero-nodal-circle axial modes below 1   kHz. The latter have radiation indices of −1.7, −18.5, −7.8 and −2.6   dB for n  =   0, 1, 2, 3.

FIGURE 6-10. Radiation ratio for modes of 920   mm freight wheel at resonance frequencies. Numbers indicate n, number of nodal diameters. Δ, zero-nodal-circle axial modes; ∇, one-nodal-circle axial modes; +, radial modes

In [6.13] boundary element results were presented for a series of notional wheels of different diameters and web thicknesses, from which the dependence of the radiation ratio on various parameters could be established. This allowed simple engineering formulae to be developed to describe the radiation ratio of a wheel in terms of basic geometrical parameters that can be implemented into a wheel/rail noise prediction model.

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Ultrasonic surgical devices and procedures

M.E. Schafer , in Power Ultrasonics, 2015

21.3.3.2 Torsional

The torsional mode of vibration involves motion that twists or "rolls" relative to the axis of the probe. One way to create this motion is by using a modified coupler between the transducer and the transmission element, as shown in Figure 21.4. Because torsional mode has a different resonant frequency than longitudinal mode, this approach allows the device to operate in either mode, depending on the electrical drive frequency. Torsional motion generally provides cutting utility only if the tip is bent or shaped to provide some surface perpendicular to the torsional motion. The cutting effectiveness of torsional systems is still controlled by the same fundamental principles of volume velocity (area multiplied by perpendicular velocity) as longitudinal systems, although there can be more flexibility in the design of the torsional end effectors. Care must be taken, however, to account for unwanted stress concentrations that can develop in torsional systems.

Figure 21.4. Transducer stack with torsional coupler element.

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Case studies of tall buildings with dynamic modification devices

Alberto Lago , ... Antony Wood , in Damping Technologies for Tall Buildings, 2019

8.1.3.3 Structural system

St Francis Shangri La Place is a development of two similar residential buildings 210   m tall and approximately 38   m square in plan located in a region of typhoon winds and UBC Zone 4 seismic conditions. Each building comprises a central core, connected in the middle by diagonally reinforced coupling beams (Figs. 8.17 and 8.18). The thickness of these walls increases toward the base, to allow for increase in both axial and shear force. A supplementary moment frame provides both additional overturning resistance for the towers and gravity support for the concrete slab floor.

Figure 8.17. St Francis Shangri La Place: tower 1 location of outrigger walls and axis system.

Figure 8.18. St Francis Shangri La Place: tower 2 location of outrigger walls and axis system.

For each of the two buildings, eight outrigger walls are attached to the core approximately half way up the building (Fig. 8.19). Two dampers are attached to the end of each of these outrigger walls, i.e., a total of 16 dampers per building.

Figure 8.19. St Francis Shangri La Place: dampers outrigger location.

8.1.3.3.1 Building fundamental periods

The approximate fundamental vibration modes of the building are as follows (Fig. 8.20):

Figure 8.20. St Francis Shangri La Place: fundamental frequencies and mode shapes of tower 1.

Tower 1: 5.73 (X direction), 5.60 (Y direction) and 4.7 (torsion) seconds

Tower 2: 8.80 (X direction), 5.82 (Y direction) and 4.9 (torsion) seconds

8.1.3.3.2 Damping strategy utilized

The damped outrigger system was used for both buildings. Details found later (Section 8.1.3.4).

8.1.3.3.3 Additional damping provided by the damping system

The additional damping achieved in each direction for 100-year wind varied between 5.2% and 11.2% of critical for the two buildings and two principal directions. Note that the final level of damping used in the design uses reduced values to allow for potential inefficiencies in the system that are not captured in the analysis, such as flexibility in the damper system (Maxwell spring effect).

8.1.3.3.4 Building cost versus damping cost

This information is not available.

8.1.3.3.5 Building code

The governing code was the National Structural Code of the Philippines (NSCP, 1999), which is very similar to UBC (1997). The design was also a very early example of performance-based design in the Philippines, and reference was made to Federal Emergency Management Agency (FEMA) 356 (ASCE, 2000) during the design process.

8.1.3.3.6 Peer-reviewed project

The project was peer reviewed by an internationally renowned academic expert. Additionally, the client also employed a separate structural firm to review.

8.1.3.3.7 Design forces

This information is not available.

8.1.3.3.8 Expected performance

Wind acceleration for 10-year wind is 9   mg (without dampers 25   mg).

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Finite element method for vibration problems

Ce Zhang , ... Zongyu Chang , in Machinery Dynamics, 2022

7.3.4.3 An example

Natural frequencies and vibration modes of a spur gear tooth can be analyzed easily with FEM software, such as ANSYS. The effects of gear materials and gear size can be compared.

(1) Gear tooth model

For an involute spur gear, the tooth profile is the same along the thickness. In addition, the tooth is symmetrical as shown in Fig. 7.13(a). Thus, this analysis can be treated as a plane-stress problem.

Figure 7.13. Mesh a spur gear tooth for natural frequency analysis.

In the profile, AB and JK are involute sections, while BC and IJ are the transitions at the root. To simplify the problem, only one tooth can be analyzed, and the tooth can be modeled as fix-ended with the boundary of $DEFGH$ . Some dimensions involved are $DE=GH=1.5$ m and, $EF=FG=2.2$ m. m here is the module of the gear. The tooth is meshed as Fig. 7.13(b). Isoparametric elements of curved sides are used. The total number of element is 465 with 493 nodes.

(2) Analysis results

Two types of materials, carbon steel, and terylene are analyzed. The gear parameter and material properties are given in Table 7.1.

Table 7.1. Gear and material properties.

Material	N	m (mm)	E (GPa)	ρ ( ${\mathrm{N}/\mathrm{m}}^2$ )	μ
steel	30	2.5	195	75,244	0.3
terylene 500	30	2.5	2.41	13,818	0.38

N: tooth number; m: module; E: Young's modulus; ρ: material density; μ: Poisson ratio.

The first four natural frequencies of the analysis are given in Table 7.2 and the corresponding vibration modes are shown in Fig. 7.14.

Table 7.2. Order of vibration modes.

	Order of vibration modes
	1	2	3	4
steel	0.781 × 105	1.631 × 105	2.203 × 105	3.401 × 105
terylene 500	0.214 × 105	0.443 × 105	0.519 × 105	0.859 × 105

Figure 7.14. Vibration modes of the gear tooth.

In Table 7.3, the primary natural frequencies of a steel gear for different gear modules are given. It is clearly seen that the smaller the gear module the lower the natural frequency.

Table 7.3. Primary natural frequency of steel gear (Hz).

Module (mm)	1	2.5	3	5
primary natural frequency	1.97 × 105	7.81 × 104	6.51 × 104	3.94 × 104

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Ultrasonic cutting for surgical applications

M. Lucas , A. Mathieson , in Power Ultrasonics, 2015

23.6.2 Multiple-mode devices

Coupling two or more modes of vibration, especially L-F modes or L-T modes, has enhanced the performance of devices used in both clinical and industrial applications and has facilitated the development of ultrasonic motors. Various design methodologies can be adopted to couple modes of vibration in ultrasonic devices. One method of generating a combined L-F mode is incorporating two piezoceramic stacks in a Langevin transducer (Figure 23.3a) and exciting them simultaneously. Set I, located at or close to the nodal plane, consists of complete, axially poled, piezoceramic rings to induce longitudinal vibration, whereas set II, located at a position that is optimal to induce flexural motion, contains two pairs of oppositely poled half-ring elements that oscillate 180° out of phase with each other (Watanabe et al., 1993 ). A combined L-T mode of vibration can also be generated by a transducer configuration similar to that exhibited in Figure 23.3a, although the stack that generates the torsional motion would be poled transversely (Ohnishi et al., 1993; Watanabe et al., 1993).

Figure 23.3. (a) Schematic of a transducer with multiple piezo stacks and (b) L-T Langevin transducer containing diagonal slots.

The approach of using multiple piezoceramic stacks to generate combinations of vibrational motions has the advantage that a high level of coupled motion can be generated at the output end of the device. However, these devices also have limitations related to the complexity of the design and difficulties with driving at multiple resonant frequencies simultaneously. An alternative method of generating combined L-F or L-T motion is to use a conventional Langevin transducer, exciting longitudinal motion, combined with an attached waveguide designed to degenerate part of the longitudinal motion into flexural or torsional motion. Ultrasonic scaling devices, used for periodontal cleaning, generally operate in a longitudinal mode; however, by incorporating a curved tip, a combined L-F motion can be excited in the scaling tool. The shape and motion of the tip permits dental clinicians to access (although to a limited extent) interproximal and other sites (Mathieson, 2012). Waveguide geometry can also be used to generate L-T motion in ultrasonic devices through the incorporation of helical or diagonal slots around an uncut core (Figure 23.3b). The longitudinal wave propagating through the uncut core will continue, while the slots deflect part of the wave to degenerate the longitudinal motion into torsional motion, resulting in combined L-T motion at the output end of the waveguide (Al-Budairi et al., 2013).

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Discrete Systems

In Modelling of Mechanical Systems, 2004

6.1 Introduction

Restricting the analysis to the linear domain, equilibrium of discrete mechanical systems having N > 1 degrees of freedom, (also called multi degrees of freedom systems, or briefly MDOF systems) is governed by a system of N linear equations, algebraic in the case of statics and differential with respect to time in the case of dynamics. In most instances, such equations couple together several generalized displacements. Therefore, it is extremely useful to find a systematic procedure allowing one to uncouple such systems of equations, in statics as well as in dynamics. Indeed, if suitably uncoupled, the system is reduced to a set of N oscillators independent of each other, which is very convenient for further analysis. The problem of uncoupling will thus serve as a guideline for most of the considerations which shall be made in this chapter.

In the first instance, section 6.2 is concerned with the task of linearizing Lagrange's equations about a static state of stable equilibrium. In the conservative case, after linearization, the system is thus characterized by a stiffness matrix [K] and a mass matrix [M]. Both of them are symmetrical, [M] is positive definite and [K] is positive.

Section 6.3 constitutes the core of the present chapter. It deals with autonomous conservative and linear systems which vibrate freely about a static state of stable equilibrium. We shall show that a judicious transformation of displacement variables allows one to uncouple the equations of motion. Moreover, the same uncoupling procedure applies also in the case of statics. The column vectors of the transformation matrix are formed by N linearly independent mode shapes of vibration of the system. As expressed on this modal basis the dynamic equations of the mechanical system reduce to a set of N uncoupled linear oscillators. The natural frequencies of such oscillators are the modal frequencies, of the system, whereas the masses and stiffness coefficients are called the modal mass and stiffness coefficients (or the generalized mass and stiffness coefficients). Clearly, the coupling between the variables on the physical basis (or in any non modal basis) is accounted for by the mode shapes which interrelate the displacement of each oscillator.

Section 6.4 extends the concept of vibration modes to systems other than those already considered in section 6.2. Namely, the following items are discussed:

1.

Natural modes of vibration of constrained systems. After linearization of the constraint conditions, the Lagrangian of a constrained system gives rise to a set of linear differential equations, mixing the variables of displacements and constraint reactions. In the conservative case, such systems can be characterized by a [K] and a [M] matrix, which are still symmetrical. However, as the variables of the problem now mix components of displacement and of forces, the physical meaning of stiffness and mass matrices [K] and [M] does not hold anymore. Nevertheless, modal analysis of such systems can proceed in the same way as in the unconstrained case.

2.

The free modes of rigid bodies, of which the natural frequency is zero. They correspond to motion allowing the potential energy of the system to remain unchanged.

3.

The modes of elastic buckling, which may occur in many prestressed systems. Such modes are analysed successively from the static and dynamical point of views.

4.

The whirling modes, which take place in rotating systems as a consequence of gyroscopic coupling (cf. Chapter 4, subsection 4.3.2).

On the other hand, extension to nonconservative systems will be described in Volumes 3 and 4, in relation to fluid-structure coupled systems. Indeed, at this stage, we will be able to bring the physical meaning of such modes to light by discussing the behaviour of a few examples.

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NORMAL MODES OF VIBRATION OF THE VERTEBRAL COLUMN†

M.M. Pant , B.Y. Tong , in Proceedings of the Sixth New England Bioengineering Conference, 1978

Publisher Summary

This chapter discusses the normal modes of vibration of the vertebral column. Interest in the structure and function of the human vertebral column ranges from its importance in evolution to the more practical aspect of spinal injuries in impulsive ejections or collisions. Damage to the vertebral column may also be caused as a result of repeated low frequency vibrations that correspond to normal mode frequencies. Calculations of the bending vibration frequencies using a simple beam theory were reported by Huijens. For long bones, a finite element method was developed in detail by Orne and Young, and applied to the dog's radius. In the case of the vertebral column, the 24 vertebrae provide a natural set of finite elements in terms of which the system should be studied. The chapter presents a series of models for the vertebral column, in a gradually increasing order of sophistication. It describes the redoing of the simple beam theory by calculating the effective bending stiffness of a disc by making use of the actual cross-sectional areas and Young's moduli as experimentally measured and reported by Yamada.

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