Normal Modes

9 Key excerpts on "Normal Modes"

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  An Introduction to Mechanics

  • Daniel Kleppner, Robert Kolenkow(Authors)
  • 2013(Publication Date)
  • Cambridge University Press

    (Publisher)

  There is always a normal mode with zero frequency, corresponding to the trivial case of a stationary system with no vibrations, leaving N − 1 non-trivial frequencies. To show an example of a normal mode, and to keep the algebra sim-ple, we go back to the diatomic molecule model with masses m 1 and m 2 . There are two equations of motion, so we expect one non-trivial vibration frequency. Using x i = a i sin( ω t + φ ) ¨ x i = − ω 2 a i sin( ω t + φ ) in Eqs. ( 1a ) and ( 1b ) gives ( k − ω 2 m 1 ) a 1 = ka 2 (2a) ( k − ω 2 m 2 ) a 2 = ka 1 . (2b) The factor sin( ω t + φ ) occurs in every term and cancels throughout. Solving Eq. ( 2a ) for a 2 and substituting in Eq. ( 2b ) gives the following equation for the amplitude a 1 , after some simplification: ω 2 [ ω 2 m 1 m 2 − k ( m 1 + m 2 )] a 1 = 0 . (3) There are several possible solutions for Eq. ( 3 ). One solution is a 1 = 0; it follows from Eq. ( 2a ) that a 2 = 0 also. This is a trivial solution for a stationary non-vibrating system. Similarly, the solution ω 2 = 0 224 TOPICS IN DYNAMICS also corresponds to a non-vibrating system. The interesting non-trivial solution is ω 2 = k ( m 1 + m 2 ) m 1 m 2 = k μ , as we expect. Now that we have the normal mode frequency, we can solve Eq. ( 2a ) for the relative amplitudes a 2 / a 1 : a 2 a 1 = − m 1 m 2 . The two masses move in opposite directions, with amplitudes that en-sure conservation of linear momentum. The actual amplitudes depend on the initial conditions. Treating a polyatomic molecule model follows the same lines, with more complicated algebra, as we shall see in the following example. Nonetheless, the final result is again the normal mode frequencies ω i in terms of the masses and spring constants. The important point is that the ω i are the only possible non-trivial vibrational frequencies of the system, and any possible motion is therefore a linear combination of the Normal Modes N − 1 i = 1 A i sin( ω i t + φ i ) .

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  Numerical Modeling of Ocean Dynamics

  • Zygmunt Kowalik, T S Murty(Authors)
  • 1993(Publication Date)
  • WSPC

    (Publisher)

  CHAPTER V

  Normal Modes

  1. Introduction

  Any water body (either completely closed or partially open) undergoes natural or free oscillations which are referred to as Normal Modes. Several different physical phenomena can set a water body into oscillation, i.e., excite its Normal Modes. The frequencies of the fundamental normal mode and its higher harmonics can be determined solely from a knowledge of the geometry of the water body and the water depths. The question of Normal Modes was first discussed in connection with tidal theories (LaPlace, 1775, 1776; Hough, 1898).

  Consider an artificial situation in which a thin layer of water covers the Earth's surface entirely. We ask the question how this water can move freely, subject to gravity and the earth's rotation. Hough (1898) showed that the free motion can occur in either of two ways. Oscillations of the first class (OFC) are essentially gravity waves whose periods are modified by the Earth's rotation. However, OFC can exist even if the Earth does not rotate. Oscillations of the second class (OSC) owe their very existence to the Earth's rotation and have periods greater than 24 h. If the earth's rotation tends to zero, OSC will lose their periodicity and will degenerate into steady currents. An example of OSC is the Rossby wave.

  If σ is the frequency of oscillation and ω is the frequency of rotation, then oscillations of the first class are those for which σ → σ0 (≠ 0) as ω → 0 and oscillations of the second class are those for which σ → O(ω) as ω → 0. Bjerknes et al. (1934) distinguished between these two types of oscillation by means of the ratio σ/2ω. Gravity modes (oscillations of the first class) are those for which σ/2ω ≥ 1. Elastoid-inertia modes (oscillations of the second class) are those for which σ/2ω ≤ 1. For the gravity modes, gravity appears in the frequency equation. In the case of the rotational (elastoid-inertia) modes, the frequency for a given mode is a function mainly of the ratio of the depth of the liquid to the radius of the container and gravity does not play an important role in the frequency equation. Here this discussion is restricted to gravity modes. In the mathematical analysis this restriction is imposed by introducing the approximations of the shallow water theory called the quasi-static approximation (Bjerknes et al.

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  Vibrations and Waves

  • A.P. French(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  2 Nevertheless, the important feature remains—that a given one-dimensional system, with specified boundary conditions, has a denumerable set (even if infinite) of characteristic natural modes of vibration.

  1 For further discussion, refer back to Chapter 5 .

  2 Even in a one-dimensional system that is treated as being continuous in structure, the relation between the mode frequencies becomes quite different if the system is not uniform—e.g., a stretched string whose thickness increases continuously from one end to the other, or (as shown in Fig. 5-1 ) a uniform chain hanging vertically, in which the tension decreases steadily from the top downward.

  Let us end this general discussion of the Normal Modes by drawing attention to two features, already referred to, that are of especially great importance:

  1. The boundary conditions , as applied in this case at the two ends of the one-dimensional system, play a decisive role in determining the character of the Normal Modes.
  2. Given the linearity of our basic equations of motion, any or all of the Normal Modes of vibration can coexist with arbitrary relative values of amplitude and phase.

  Normal Modes of a Two-Dimensional System

  We shall turn our attention now to a brief consideration of the Normal Modes of systems that are essentially two-dimensional, such as a stretched elastic sheet or a thin metal plate. As with the one-dimensional systems, the specification of boundary conditions —now, primarily, around the edges—limits the permissible motions to a few particular classes: the Normal Modes that are consistent with the stated boundary conditions. The precise character of the Normal Modes may be very beautifully indicative of any symmetries that a given physical system possesses.

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  An Introduction to Groups and their Matrices for Science Students

  • Robert Kolenkow(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  (3.5) can be solved for the c ij , which gives c 12 D c 11 for normal mode 1 and c 21 D c 22 for normal mode 2. From Eq. (3.2): x 0 1 D c 11 x 1 C c 12 x 2 D c 11 .x 1 C x 2 / x 0 2 D c 21 x 1 C c 22 x 2 D c 22 .x 1 C x 2 /: In the language of group theory, x 1 C x 2 and x 1  x 2 are basis functions for the two modes. This will become clearer in Section 3.3 where Normal Modes are treated according to group theory. Assume that the masses are at maximum amplitude in their motion. To visualize the motion in normal mode 1, note that both terms in x 1 C x 2 have the same sign – the two masses are moving together in phase as the sketch indi- cates. In mode 2, x 1 and x 2 have opposite signs so that the masses move in opposite directions 180 ı out of phase. Perhaps you can see qual- itatively from the sketches why mode 1 has a lower oscillation frequency than mode 2. Such mode diagrams are often used in the discussion of molecular vibrations. The masses are constantly in motion, and half a period later the amplitudes will again be at maximum but reversed as shown in the sketch. 3.3 Normal Modes and Group Theory An oscillating system has Normal Modes n that satisfy equations for simple harmonic motion. R x 0 n C ! 2 n x 0 n D 0 (3.8) The object of this section is to show that the Normal Modes of an oscillating system are in fact the basis functions of the irreducible representations of the system’s symmetry group. 64 3 Molecular Vibrations Suppose that the oscillating system being considered has symmetry properties described by a symmetry group with members T. This idea will be key for the discus- sion of molecular vibrations in Section 3.4. Suppose further that the Normal Modes x 0 are basis functions for the group so that they generate a matrix representation with matrices D 0 .T/. Tx 0 n D X j D 0 jn .T/x 0 j T R x 0 n D X j D 0 jn .T/ R x 0 j D  X j D 0 jn .T/! 2 j x 0 j (3.9) using the equation of motion Eq.

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  Introduction to Seismology

  • Peter M. Shearer(Author)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  The string will resonate only at certain fre- quencies (Figure 8.15). These are termed the standing waves for the string and any motion of the string can be expressed as a weighted sum of the standing waves. This is an eigenvalue problem; the resonant frequencies are termed the eigen- frequencies; the string displacements are termed the eigenfunctions. In a musical instrument the lowest frequency is called the fundamental mode; the higher modes are the overtones or harmonics. For the vibrating string, the eigenfunctions are sines and cosines and it is natural to use a Fourier representation. Normal Modes for the Earth are also specified by their eigenfrequencies and eigenfunctions. A detailed treatment of normal mode theory for the Earth is beyond the scope of this book, and computation of eigensolutions for realistic Earth models is a formidable task. However, it is useful to remember some of the properties of the eigenfunctions of any vibrating system: 1. They are complete. Any wave motion within the Earth may be expressed as a sum of Normal Modes with different excitation factors. 2. They are orthogonal in the sense that the integral over the volume of the Earth of the product of any two eigenfunctions is zero. This implies that the normal mode representation of wave motion is unique. What do Earth’s Normal Modes look like? For a spherically symmetric solid, it can be shown that there are two distinctly different types of modes: spheroidal modes, which are analogous to P-SV and Rayleigh wave motion, and toroidal modes, which are analogous to SH and Love wave motion. The Earth’s departures from spherical symmetry mean that this separation is not complete, but it is a very good first-order approximation. Toroidal modes involve no radial motion and are 228 8 Surface Waves and Normal Modes sensitive only to the shear velocity, whereas spheroidal modes have both radial and horizontal motion and are sensitive to both compressional and shear velocities.

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  Advanced Theoretical Mechanics

  A Course of Mathematics for Engineers and Scientists

  • Brian H. Chirgwin, Charles Plumpton(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  The system rests on a smooth hori-zontal table and the particles can oscillate transversely. Show that the squares of the reciprocals of the periods of the normal oscillations are in the ratios 2 -]/2 :2 :2 + ]/2. Sketch the typical configuration in each of the three Normal Modes. 6. One end of a uniform rod of mass 3 m and length 3 u s pivoted to a fixed point and to the other end is attached a light elastic string of natural length L — L', which carries a mass m at its end. In the equilibrium position, the length of the string is L. Show that the small oscillations of the system in a vertical plane have periods of simple pendulums of lengths L 3L, 2L/5. Find the normal coordinates. 7. A thin uniform square plate of side 2a is suspended by two light inextensible strings, each of length 4a/3, which are attached to the ends A, B of an edge and to two points C, D at the same level at a distance 2 a apart. Describe the Normal Modes of oscillation and prove that the lengths of the equivalent simple pendulums are 4a/3, 4a/9, 2(2 ± ]/3) a/3. 8. A chain of n + 1 equal light springs, each of unstretched length I and strength λ (i.e. X(L — I) is the force in the spring when its length is L), connects two fixed points a distance (n + 1)1 apart and in the same vertical line. Each junction of two springs is loaded with a particle of mass m. Show that in equilibrium the distance of the kth. particle from the top is mq Jcl + —L]c(n+ 1 -k). 2λ The system executes small oscillations in the vertical line joining the two fixed points. Show that the circular frequencies ω of the Normal Modes are given by the vanishing of the n x n determinant D n = 2cos0 -1 0 -1 2cos0 -1 0 -1 2cos0 0 0 -1 . 0 0 0 0 where cos0 = 1 — meo 2 /2A. 0 -1 2cos0 -1 0 0 -1 2cos0| [see over] 376 A COURSE OF MATHEMATICS By establishing a recurrence relation for D n , or otherwise, prove that D n = sin(?i + l)0/sin0 and thus find the frequencies of the Normal Modes.

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  Quantum Chemistry

  An Introduction

  • Walter Kauzmann(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  The lines separating the -f and -regions are nodal lines (where the membrane is at rest at all times). Degenerate Normal Modes can also give interesting shapes if they are combined with different phases. The nodal lines are then no longer stationary, but move about on the vibrating object. For instance, let us superimpose the modes / 1 2 and / 2 1 of the square membrane, letting them be one quarter cycle out of phase with each other, so that z (x ,y,t) = f 2i (x ,y)cos{2ntIT 0) - fi 2 (x ,y )sm (2ntlTo) where To is the period of the oscillation. The nodal line will then move about on the membrane in the manner shown in Fig. 3-11. These motions are, of course, repeated every To seconds. It is evident, then, that when there is degeneracy one can construct an infinite number of Normal Modes having different shapes, but the same frequency. Of these only a small number are really independent modes, however. The degeneracy is in fact defined as the number of linearly in- 76 VIBRATION THEORY-TYPICAL SYSTEMS dependent modes of a given frequency; that is, the largest number of modes for which it is not possible to find a set of constants c± t c

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  Fundamentals of Sound and Vibration

  • Frank Fahy, David Thompson, Frank Fahy, David Thompson(Authors)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  The free and forced vibrations can be written as a sum of modal terms; and, of course, each mode behaves like an SDOF system. 3.7.3 Waves in structures Structural vibration can be represented in terms of waves as well as modes. In fact, the modes of distributed systems are simply the inter-ference patterns (standing waves) formed by waves travelling around the structure. The wave interpretation is particularly useful at higher frequencies (e.g. higher than the first few natural frequencies) when we might want to describe the transmission of vibration from a source, such as a machine, through the structure to which the source is attached, to some receiver. For the case of axial vibration in a rod, we note that we could also write the solution to the equation of motion, Equation 3.137, as U x B C kx kx ( ) = + -e e i i so that the axial displacement is u x t B C t kx t kx , ( ) = + -( ) + ( ) e e i i w w (3.148) The two terms represent waves which propagate in the positive and nega-tive x -directions respectively. These are analogous to sound waves in air. In Table 3.4 Natural frequencies and mode shapes for rods Boundary conditions Characteristic equation Mode shapes Natural frequencies cos w L c = 0 sin 2 1 2 n x L + ( ) π 2 1 n c L -( ) π sin w L c = 0 cos n x L π n c L π a sin w L c = 0 cos n x L π n c L π Note: Boundary conditions fixed ( u = 0) and free ( P = 0). a Existence of rigid body modes. Fundamentals of vibration 139 Equation 3.148, k is the wavenumber (spatial frequency or phase change per unit distance), while the wavelength is λ = 2 π / k . The waves travel with a speed c k E = = w r . The situation is different for bending waves (Equation 3.144).

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  Fundamentals of Noise and Vibration Analysis for Engineers

  • M. P. Norton, D. G. Karczub(Authors)
  • 2003(Publication Date)
  • Cambridge University Press

    (Publisher)

  They represent a set of co-ordinates which are orthogonal to each other. Each principal co-ordinate, q n , thus gives the relative amplitude of displacement, velocity and acceleration of the total system at a given natural frequency, ω n , and the linear sum of all the principal co-ordinates gives the total response. The concepts of principal co-ordinates are used in the normal mode vibration analysis of continuous structures, and this is discussed in the next section. When damping is considered, a damping matrix, C v , has to be included in the equa-tions of motion. In general, the introduction of damping couples the equations of motion because the off-diagonal terms in the damping matrix are not zero – i.e. coupled sets of ordinary, differential equations result. Often, because damping is generally small in mechanical and structural systems, approximate solutions are obtained by considering 64 1 Mechanical vibrations the coupling due to damping to be a second order, i.e. c v i j c v j j for i = j . Techniques for the modal analysis of damped, multiple-degree-of-freedom systems are described in many texts on mechanical vibrations (e.g. Tse et al . 1 . 5 ). 1.9 Continuous systems – a review of wave-types in strings, bars and plates At the very beginning of this book it was pointed out that engineers tend to think of vibrations in terms of modes and of noise in terms of waves, and that quite often it is forgotten that the two are simply different ways of looking at the same physical phenomenon! When considering the interactions between noise and vibration, it is important for engineers to have a working knowledge of both physical models. Any continuous system, such as an aircraft structure, a pipeline, or a ship’s hull, has its masses and elastic forces continuously distributed (as opposed to the rigid masses and massless springs discussed in previous sections).

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