mode shapes & natfr_details

8/13/2019 Mode Shapes & Natfr_details

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Mendelsohn, D. A. Free Vibration, Natural Frequencies, and Mode ShapesThe Engineering Handbook.

Ed. Richard C. Dorf

Boca Raton: CRC Press LLC, 2000

1998 by CRC PRESS LLC


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14Free Vibration, Natural Frequencies,

14.1 Basic Principles

14.2 Single-Degree-of-Freedom SystemsEquation of Motion and Fundamental Frequency Linear Damping

14.3 Multiple-Degree-of-Freedom Systems

14.4 Continuous Systems (Infinite DOF)

Daniel A. MendelsohnOhio State University

14.1 Basic Principles

In its simplest form, mechanical vibration is the process of a mass traveling back and forth

through its position of static equilibrium under the action of a restoring forceor momentthat

tends to return the mass to its equilibrium position. The most common restoring mechanismis a spring or elastic member that exerts a force proportional to the displacement of the mass.

Gravity may also provide the restoring action, as in the case of a pendulum. The restoring

mechanism of structural members is provided by the elasticity of the material of which the

member is made. Free vibrationis a condition in which there are no external forces on the

system.

Cyclicorperiodicmotion in time is described by the property x(t + ) = x(t), where tistime and is theperiod,that is, the time to complete one cycle of motion. The cyclicfrequencyof the motion is f = 1=, usually measured in cycles per second (Hz). The

special case of periodic motion shown inFig. 14.1is harmonic motion,

x(t) = A sin(! t) + Bcos(! t) (14:1a)

= X sin(! t + ) (14:1b)

where ! = 2fis the circular frequency,typically measured in radians/s,

andMode Shapes



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X = (A2 + B 2)1=2 is the amplitudeof the motion, and = tan 1(B =A)is thephase angle.Many systems exhibit harmonic motion when in free vibration, but do so only at discrete

natural frequencies.A vibrating system with n degrees of freedom(DOF) has nnatural

frequencies, and for each natural frequency there is a relationship between the amplitudes of

the nindependent motions, known as the mode shape.A structural elastic member has an

infinite number of discrete natural frequencies and corresponding mode shapes. The

fundamental frequencyand associated mode shape refer to the smallest natural frequency

and associated mode shape. The study of free vibrations consists of the determination of the

natural frequencies and mode shapes of a vibrating system as a function of geometry,

boundary conditions, mass (density) of the components, and the strength of the restoring

forces or moments. Although the natural frequencies and mode shapes are valuable to know

by themselves, they have perhaps their greatest value in the analysis of forced vibrations, as

discussed in detail in the following chapter.

Figure 14.1 Time history of undamped periodic or cyclic motion.



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Equation of Motion and Fundamental Frequency

The system shown inFig. 14.2(a)consists of a mass, m, that rolls smoothly on a rigid floor

and is attached to a linear spring of stiffness k. Throughout this chapter all linear (or

longitudinal) springs have stiffnesses of dimension force per unit change in length fromequilibrium, and all rotational (or torsional) springs have stiffnesses of dimension moment

per radian of rotation from equilibrium (i.e., force times length). The distance of the mass

from its equilibrium position, defined by zero stretch in the spring, is denoted byx. Applying

Newton's second law to the mass inFig. 14.3(a)gives the equation of motion:

kx= md2xdt2

) md2xdt2

+ kx= 0 (14:2)

Alternatively, Lagrange's equation (with only one generalized coordinate,x) may be used to

find the equation of motion:

ddt

@L

@(dx=dt)

@L

@x = 0 (14:3)

The Lagrangian,L, is the difference between the kinetic energy, T, and thepotential energy,

U, of the system. The Lagrangian for the system inFig. 14.2(a)is

L T U=

1

2m

dx

dt

2

1

2kx2

(14:4)

Substituting Eq. (14.4) into Eq. (14.3) gives the same equation of motion as in Eq. (14.2).

Using the harmonic form in Eq. (14.1) forx, Eq. (14.2) is satisfied if ! takes on the value

! =p

k=m (14:5)

which is therefore the natural frequency of the system. If the displacement and velocity are

known at some time (say, t = 0), then the constants in Eq. (14.1) may also be evaluated,

A = 1

! 0

dxdt

(0); B = x(0) (14:6)

and the corresponding displacement history is shown inFig. 14.1.

14.2 Single - Degree - of Freedom Systems



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damping, and (c) with frictional damping.

Figure 14.3 Free-body diagrams of the single-degree-of-freedom systems in Figure 14.2.

The natural frequency for conservative systems can also be found by the energy method.

As the mass passes through equilibrium, U = 0andT =Tmax , while at its maximumdisplacement where the mass has zero velocity,T = 0and U = Umax . Since the total energyis constant, ! is the frequency for whichTmax = Umax . Using Eq. (14.1a) and the system inFig. 14.2(a),this principle gives

Tmax =12

m(! X )2 =12

kX 2 = Umax (14:7)

which in turn gives the same result for ! as in Eq. (14.5).Table 14.1contains the equation of motion and natural frequency for some single-DOF

systems. Gravity acts down and displacements or rotations are with respect to static

equilibrium. The mode shapes are of the form in Eq. (14.1) with ! given inTable 14.1.

Figure 14.2 Typical one-degree-of-freedom system: (a) without damping, (b) with viscous



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Table 14.1 Equations of Motion and Natural Frequencies for some Single-DOF Systems



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Linear Damping

Figures14.2(b)and 14.3(b)show an example of viscous damping caused by a dashpot of

strength c(force per unit velocity) that acts opposite the velocity. Newton's second law then

gives

kx cdxdt

= md2xdt2

) md2xdt2

+ cdxdt

+ kx= 0 (14:8)

which has the solution

x(t) = e ! 0t [A sin(! dt) + Bcos(! dt)]

= X e ! 0t sin(! dt + )(14:9)

where the damped natural frequency, ! d, damping factor, , and critical damping coefficient,cc, are given by

! d !p

1 2; ccc

; cc 2m! = 2p

mk (14:10)

respectively. If c < cc, then ! dis real and Eq. (14.9) represents exponentially dampedoscillation, as shown inFig. 14.4. If c cc, then the system is supercritically dampedanddecaying motion but no vibration occurs.

Figure 14.4 Time history of viscously

damped vibration.Figure 14.5 Time history of frictionally

damped vibration.



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14.3 Multiple-Degree-of-Freedom Systems

For each DOF in an n-DOF system there is a coordinate, xi (i = 1; 2; : : : ; n), which is a

measure of one of the independent components of motion. The motion of the system isgoverned by n, generally coupled, equations of motion, which may be obtained by a

Newtonian approach requiring complete free-body and acceleration diagrams for each mass.

For systems with many DOFs this approach becomes very tedious. Alternatively, applying

Lagrange's equations, with no damping present,

ddt

@L@_xi

@L@xi

= 0; (i = 1; 2; : : : ; n) (14:13)

to the particular system yields the nequations of motion in the nunknown coordinates xi ,

[M ]

d2xdt2

+ [K]fxg= f0g (14:14)

The frictional effects inFigs.14.2(c)and14.3(c)are characterized by a Coulomb frictional

force F = N = mg, where is the coefficient of sliding friction. The equation of motionis then

md2x

dt2 + ( mg)sgn dx

dt

+ kx= 0 (14:11)

where sgn(dx=dt)is equal to +1or 1for positive or negative values of dx=dt,respectively. This equation must be solved separately for each nth half period of the

oscillation of frequency, ! ,

x(t) = [x0 (2n 1)]cos(! t) sgn

dxdt

(14:12)

where = ( mg=k)is the minimum initial displacement to allow motion, and ! is theundamped natural frequency, Eq. (14.5).Figure 14.5shows x(t)for an initial displacementof x0= 20 .



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As stated before, the Lagrangian, L =T U , is the difference between the kinetic andpotential energies of the system. [M ]and [K]are the n nmassand stiffness matrices,withelements mij and kij , which multiply the acceleration and displacement vectors of themasses, respectively. Writing xi in the form of Eq. (14.1), Eq. (14.14) yields nhomogeneousequations, [A]fX g= f0g, in the namplitudes X i . The elements of [A]are

aij = kij mij !

2

. If a solution exists, the determinant of [A], an nth order polynomial in !

2

,must be zero. This yields thefrequencyor characteristic equation,whose nroots are the

natural frequencies squared, (! i )2. Each mode shape may be written as a vector of n 1amplitude ratios:

X 2X 1

;X 3X 1

; : : : ;X nX 1

(i)

; (i = 1; 2; : : : ;n) (14:15)

The ratios are found by eliminating one equation of [A]fX g= f0g, dividing the remainingn 1equations by X 1, and solving. Then setting ! = ! i gives the ith mode shape.Equations of motion, natural frequencies, and mode shapes for some two-DOF systemsundergoing small amplitude vibrations are in Table 14.2. Gravity acts down and

displacements and rotations are taken with respect to the position of static equilibrium.

Table 14.2 Equations of Motion, Natural Frequencies, and Mode Shapes for some Two-DOF Systems



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14.4 Continuous Systems (Infinite DOF)

The equations of motion of structural members made up of continuously distributed elastic or

flexible materials are most easily obtained by a Newtonian analysis of a representative

volume element. As an example, consider the longitudinal vibration of an elastic rod

(Young's modulusE, density ) of cross-sectional areaA. A free-body diagram of a volumeelement A dx, with normal stresses [x](x)and [x+ (@x=@x)dx](x)acting on the crosssections is shown inFig. 14.6(a).A circular section is shown but the analysis applies to any

shape of cross section. If u(x)is the displacement in thexdirection of the cross section atx,then Newton's second law gives

xA+

x+@x

@x dx

A = (A dx)@2u@t2

(14:16)

Simplifying, letting dxgo to zero, and noting uniaxial Hooke's law and the definition of thestrain, "x ,



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x = E "x = E@u@x

(14:17)

Eq. (14.16) can be written as

@2

u@x2 =

E

@2

u@t2 (14:18)

which is the equation of motion for standing modes of free vibration and for wave

propagation along the rod at velocity c =p

E =. If u(x;t) = U(x)[A sin(! t) + Bcos(! t)],then Eq. (14.18) gives

d2Udx2

2U = 0; 2 =! 2

E (14:19)

which has solution U(x) = Csin(x) + Dcos(x) . Now as an example, consider thefixed-fixed bar of lengthLshown in Fig. 14.6(b)that has boundary conditions (BCs)

U(0) = 0and U(L ) = 0, which give, respectively, D = 0and either C = 0, which is not ofinterest, or

sin(L ) = 0 ) = n =n

L ) ! n =

nL

sE

(n = 1; 2;3; : : :) (14:20)

This is the frequency equation and the resulting infinite set of discrete natural frequencies for

the fixed-fixed beam of lengthL. The mode shapes are Un(x) = sin(nx).

Figure 14.6 Longitudinal vibration of a rod of circular cross section: (a) free-body diagram ofrepresentative volume element and (b) a clamped-clamped rod of lengthL.

The transverse motion y(x;t)of a taut flexible string (tension Tand mass per unit length), the longitudinal motion u(x;t)of a rod (Young's modulusE), and the torsional rotation(x;t)of a rod of circular or annular cross section (shear modulus G) all share the samegoverning equations (14.18) and (14.19), but with different values: 2 = ! 2=T , ! 2=E ,and ! 2=G, respectively.Tables 14.3and14.4contain frequency equations, nondimensionalnatural frequencies, and mode shapes for various combinations of BCs for a rod of length L.

Only the fixed-fixed conditions apply to the string.



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Table 14.4 Nondimensional Natural Frequencies (nL )1

Table 14.3 Longitudinal and Torsional Vibration of a Rod

| |

n 0 .5 1 2 5 10 100

For Longitudinal and Torsional Clamped/Spring and Free/Mass BCs2

1 2.289 2.029 1.837 1.689 1.632 1.577 /2

2 2 5.087 4.913 4.814 4.754 4.734 4.715 3/2

3 3 8.096 7.979 7.917 7.879 7.867 7.855 5/2

For Longitudinal and Torsional Clamped/Mass and Torsional Free/Spring BCs3

1 0 .653 .860 1.077 1.314 1.429 1.555 /2

2 3.292 3.426 3.644 4.034 4.306 4.666 3/2

3 2 6.362 6.437 6.578 6.910 7.228 7.776 5/2

For Longitudinal Free/Spring BCs4

1 2.975 2.798 2.459 1.941 1.743 1.587 /2

2 2 6.203 6.121 5.954 5.550 5.191 4.760 3/2

3 3 9.371 9.318 9.211 8.414 8.562 7.933 5/2



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1For the nonclassical boundary conditions in Table 14.3.2SeeTable 14.3,cases 1 and 4.3See Table 14.3,cases 2 and 3.

4See Table 14.3,case 3.

The transverse deflection of a beam, w(x;t), is governed by the equation of motion,

@4w@x4

=

AE I

@2w@t2

(14:21)

which, upon substitution of w(x;t) = W(x)[A sin(! t) + Bcos(! t)], leads to

d4w

dx4 4

W = 0; 4

=

A! 2

E I (14:22)

This equation has the general solution

W(x) = c1 sin(x) + c2 cos(x) + c3 sinh(x) + c4 cosh(x) . The frequency equation,natural frequencies, and normalized mode shapes are found by applying the BCs in the same

manner as above. The results for various combinations of simply supported (SS:

W = W00 = 0), clamped (C: W = W0= 0), and free (F: W00 = W000 = 0) BCs for a beam oflengthLand flexural rigidity E I are given inTable 14.5.

Defining Terms

Cyclic and circular frequency: The cyclic frequency of any cyclic or periodic motion is the

number of cycles of motion per second. One cycle per second is called a hertz (Hz). The

circular frequency of the motion is 2times the cyclic frequency and converts one cycleof motion into 2radians of angular motion. The circular frequency is measured inradians per second.

Degree of freedom (DOF): An independent motion of a moving system. A single mass

rolling on a surface has one DOF, a system of two masses rolling on a surface has two

DOFs, and a continuous elastic structure has an infinite number of DOFs.



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Free vibration: The act of a system of masses or a structure vibrating back and forth about

its position of static equilibrium in the absence of any external forces. The vibration is

caused by the action of restoring forces internal to the system or by gravity.

Fundamental frequency: The smallest natural frequency in a system with more than one

DOF.

Mode shape: The relationship between the amplitudes (one per DOF) of the independent

motions of a system in free vibration. There is one mode shape for each natural

frequency and it depends on the value of that natural frequency. For a continuous elastic

structure the mode shapes are the shapes of the structure at its maximum deformation

during a cycle of vibration.

Natural frequency: The frequency or frequencies at which a system will undergo free

vibration. There is one natural frequency per DOF of the system. Natural frequencies

depend on the geometry, the boundary conditions (method of support or attachment),

the masses of the components, and the strength of the restoring forces or moments.

References

Clark, S. K. 1972.Dynamics of Continuous Elements.Prentice Hall, Englewood Cliffs, NJ.

Den Hartog, J. P. 1956.Mechanical Vibrations,4th ed. McGraw-Hill, New York.

Gorman, D. J. 1975. Free Vibration Analysis of Beams and Shafts.John Wiley & Sons, New

York.

Leissa, A. W. 1993a. Vibrations of Plates.Acoustical Society of America, New York.

(Originally issued by NASA, 1973.)

Leissa, A. W. 1993b. Vibrations of Shells.Acoustical Society of America, New York.

(Originally issued by NASA, 1973.)Magrab, E. B. 1979. Vibrations of Elastic Structural Members.Sijthoff and Noordhoff,

Leyden, The Netherlands.

Meirovitch, L. 1967.Analytical Methods in Vibrations.Macmillan, New York.

Thomson, W. T. 1988. Theory of Vibrations with Applications.Prentice Hall, Englewood

Cliffs, NJ.

Timoshenko, S. P., Young, D. H., and Weaver, J. W. 1974. Vibration Problems in

Engineering,4th ed. John Wiley & Sons, New York.

Further Information

There are several excellent texts that discuss the free vibrations of discrete systems (finite

number of DOFs). In particular the books by Den Hartog [1956], Timoshenko et al.[1974],

and Thomson [1988] are recommended.

Extensive data for the natural frequencies of beams having elastic supports (translational or

rotational), end masses, multiple spans, discontinuities in cross sections, axial tension or

compression, variable thickness, or elastic foundations may be found in the monograph by



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Gorman.

Other important structural elements are plates and shells. Plates are flat, whereas shells

have curvature (e.g., circular cylindrical, elliptic cylindrical, conical, spherical, ellipsoidal,

hyperboloidal). A summary of natural frequencies for plates obtained from 500 other

references is available in the book on plate vibrations by Leissa [1993a]. Extensive frequency

data for various shells taken from 1000 references is also available in the book on shell

vibrations by Leissa [1993b].

BCs Frequency Equation 1 2 Asymptotic to Normalized Mode ShapeC-F 1 + coscosh = 0 1:875 4:694 (2n+ 1)=2

(cosh nx cosnx)

n(sinh nx sinnx);

n = cosh n + cos

sinhn + sin

SS-SS sin= 0 2 n sinnxC-SS tanh tan= 0 3:927 7:069 (4n+ 1)=4

(cosh nx cosnx)

n(sinh nx sinnx);

n = cosh n cos

sinhn sinF-SS tanh tan= 0 3:927 7:069 (4n+ 1)=4

(cosh nx+ cos nx)

n(sinh nx+ sin nx);

n = cosh n cos

sinhn sinC-C 1 coscosh = 0 4:730 7:853 (2n+ 1)=2

(cosh n cos

nx)

n(sinh nx sinnx);

n = sinhn + sin

cosh n cosF-F 1 coscosh = 0 4:730 7:853 (2n+ 1)=2

(cosh n + cosnx)

n(sinh nx+ sin nx);

n = sinhn + sin

cosh n cos

Table 14.5 Transverse Vibrations of a Beam

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