Dynamics of Mechanical Systems 2009 Part 10 pot

12.3.3, we obtain the governing dynamical equations of the system: 12.5.18 12.6 Closure The computational and analytical advantages of Kane’s equations and Lagrange’s tions are illustrat

If we let the joint moments (M1, M2, and M3) be zero and if we let the point mass M

also be zero in Eqs (12.4.20), (12.4.21), and (12.4.22), we see that the equations are identical

Generalized Dynamics: Kane’s Equations and Lagrange’s Equations 433

to Eqs (8.11.1), (8.11.2), and (8.11.3) reportedly developed using d’Alembert’s principle.Although the details of that development were not presented, the use of Lagrange’sequations has the clear advantage of providing the simpler analysis As noted before, thesimplicity and efficiency of the Lagrangian analysis stem from the avoidance of theevaluation of accelerations and from the automatic elimination of nonworking constraint

forces In the following section, we outline the extension of this example to include N rods.

Consider a pendulum system composed of N identical pin-connected rods with a point mass Q at the end as in Figure 12.5.1 (we considered this system [without the end mass]

in Section 8.11) As before, let each rod have mass m and length  Also, let us restrict our

analysis to motion in the vertical plane The system then has N degrees of freedom

represented by the angles θi (i = 1,…, n) as shown in Figure 12.5.1 This system is useful

for modeling the dynamic behavior of chains and cables

We can study this system by generalizing our analysis for the triple-rod pendulum.Indeed, by examining Eqs (12.4.1) through (12.4.10), we see patterns that can readily be

generalized To this end, consider a typical rod B i as in Figure 12.5.2 Then, from Eq

(12.4.1), the angular velocity of B i in the fixed inertia frame may be expressed as:

(12.5.1)

where n3 is a unit vector normal to the plane of motion as in Figure 12.5.2

Next, from Eq (12.4.2), the velocity of the mass center G i may be expressed as:

(12.5.2)

where as before, and as in Figure 12.5.2, the niθ are unit vectors normal to the rods

Similarly, the velocity of the point mass Q is:

θ θ

From Eq (12.4.4) we see that the partial angular velocities of B i are:

where M is the mass of Q.

From a generalization of Eq (12.4.11) the kinetic energy K of the N-rod system is:

j i

G j j

n n

20

Generalized Dynamics: Kane’s Equations and Lagrange’s Equations 435

Finally, substituting from Eqs (12.5.6) and (12.5.9) into Lagrange’s equations in the form

of Eq (12.3.3), we obtain the governing dynamical equations of the system:

(12.5.18)

12.6 Closure

The computational and analytical advantages of Kane’s equations and Lagrange’s tions are illustrated by the examples In each case, the effort required to obtain thegoverning dynamical equations is significantly less than that with d’Alembert’s principle

equa-or Newton’s laws As noted earlier, the reason fequa-or the reduction in effequa-ort is that working constraint forces are automatically eliminated from the analysis with Kane’sand Lagrange’s equations; hence, an analyst can ignore such forces at the onset Also,with Kane’s and Lagrange’s equations, the exact same number of governing equationsare obtained as the degrees of freedom Finally, Lagrange’s equations offer the additionaladvantage of using energy functions, which makes the computation of vector accelera-tion unnecessary The disadvantages of Lagrange’s equations are that they are notapplicable with nonholonomic systems, and the differentiation of the energy functionsmay be tedious and even unwieldy for large systems

non-In the following chapters we will consider applications of these principles in tions, stability, balancing, and in the study of mechanical components such as gearsand cams

12.1 Kane, T R., Dynamics of nonholonomic systems, J Appl Mech., 28, 574, 1961.

12.2 Kane, T R., Dynamics, Holt, Rinehart & Winston, New York, 1968, p 177.

12.3 Huston, R L., and Passerello, C E., On Lagrange’s form of d’Alembert’s principle, Matrix

Tensor Q, 23, 109, 1973.

12.4 Papastavridis, J G., On the nonlinear Appell’s equations and the determination of generalized

reaction forces, Int J Eng Sci., 26(6), 609, 1988.

12.5 Huston, R L., Multibody dynamics: modeling and analysis methods [feature article], Appl.

Mech Rev., 44(3), 109, 1991.

12.6 Huston, R L., Multibody dynamics formulations via Kane’s equations, in Mechanics and Control

of Large Flexible Structures, J L Jenkins, Ed., Vol 129 of Progress in Aeronautics and Astronautics,

American Institute of Aeronautics and Astronautics (AIAA), 1990, p 71.

12.7 Huston, R L., and Passerello, C E., Another look at nonholonomic systems, J Appl Mech., 40,

101, 1973.

12.8 Kane, T R., and Levinson, D A., Dynamics, Theory, and Applications, McGraw-Hill, New York,

1985, p 100.

Problems

Section 12.2 Kane’s Equations

P12.2.1: Consider the rotating tube T, with a smooth interior surface, containing a particle

P with mass m, and rotating about a vertical diameter as in Problems P11.6.6 and P11.9.6 and as shown again in Figure P12.2.1 As before, let the radius of T be r, let the angular speed of T be Ω, and let P be located by the angle θ as shown in Figure P12.2.1 This

system has one degree of freedom, which may be represented by θ Use Kane’s equations,

Eq (12.2.1), to determine the governing dynamical equation

P12.2.2: Consider the pendulum consisting of a rod with length  and mass m attached

to a circular disk with radius r and mass M and supported by a frictionless pin as in

Problems P11.9.1 and P11.12.1 and as shown again in Figure P12.2.2 This system has onedegree of freedom represented by the angle θ Use Kane’s equations to determine thegoverning dynamical equation

Generalized Dynamics: Kane’s Equations and Lagrange’s Equations 437

P12.2.3: Consider the rod pinned to the vertically

rotating shaft as in Problems P11.9.2 and as shown

again in Figure P12.2.3 If the shaft S has a specified

angular speed Ω, the system has only one degree of

freedom: the angle θ between the rod B and S Use

Kane’s equations to determine the governing

dynamical equation where B has mass m and length

 Assume the radius of S is small.

P12.2.4: Repeat Problem P12.2.3 by including the

effect of the radius r of the shaft S Let the mass of S

be M.

P12.2.5: See Problems P11.9.4 and P12.2.3 Suppose

the rotation of S is not specified but instead is free,

or arbitrary, and defined by the angle φ as in Problem

P11.9.4 and as represented in Figure P12.2.5 This

system now has two degrees of freedom represented

by the angles θ and φ Use Kane’s equations to

deter-mine the governing dynamical equations, assuming

the shaft radius r is small.

P12.2.6: Repeat Problem P12.2.5 by including the

effect of the shaft radius r and the shaft mass M.

P12.2.7: Consider a generalization of the double-rod

pendulum where the rods have unequal lengths and

unequal masses as in Figure P12.2.7 Let the rod

lengths be 1 and 2, and let their masses be m1 and

m2 Let the rod orientations be defined by the angles

θ1 and θ2, as shown Assuming frictionless pins,

determine the equations of motion by using Kane’s

equations

P12.2.8: See Problem P12.2.7 Suppose an actuator (or

motor) is exerting a moment M1 at support O on the

upper bar and suppose further that an actuator at

the pin connection between the rods is exerting a

moment M2 on the lower rod by the upper rod (and

hence a moment –M2 on the upper rod by the lower

rod) Finally, let there be a concentrated mass M at

the lower end Q of the second rod, as represented in

Figure P12.2.8 Use Kane’s equations to determine

the equations of motion of this system

P12.2.9: Repeat Problems P12.2.7 and P12.2.8 using

the relative orientation angles β1 and β2, as shown in

Figure P12.2.9, to define the orientations of the rods

P12.2.10: See Problems P11.6.8 and P11.9.8 Consider

again the heavy rotating disk D supported by a light

yoke Y which in turn can rotate relative to a light

horizontal shaft S which is supported by frictionless

bearings as depicted in Figure P12.2.10 Let D have

mass m and radius r Let angular speed Ω of D in Y

be constant Let the rotation of Y relative to S be

measured by the angle β and the rotation of S in its bearings be measured by the angle α

as shown Recall that this system has two degrees of freedom, which may be represented

by the angles α and β By following the procedures outlined in Problems P11.6.8 andP11.9.8, use Kane’s equations to determine the governing equations of motion

P12.2.11: See Problems P11.6.7 and P11.9.7 Consider again the right circular cone C with altitude h and half-central angle rolling on an inclined plane as in Figure P12.2.11 As

before let the incline angle be β and let the position of C be determined by the angle φ between the contacting element of C and the plane and a line fixed in the plane as shown.

Let  be the element length of C, and let r be the base radius of C Let O be the apex of

C, and let G be the mass center of C Finally, let ψ measure the roll of C, as shown in Figure

P12.2.11 This system has one degree of freedom (see Problem P12.2.11) Use Kane’sequations to determine the equations of motion

Section 12.3 Lagrange’s Equations

P12.3.1 to P12.3.10: Repeat Problems P12.2.1 to P12.2.10 by using Lagrange’s equations toobtain the equations of motion Compare the analysis effort with that of using Kane’sequations

FIGURE P12.2.9

Double, unequal-rod pendulum with

relative orientation angles.

In this chapter we will develop a brief and elementary introduction to mechanicalvibration It is only an introduction and is not intended to replace a course or a moreintense study The reader is referred to the references, which provide a partial listing ofthe many books devoted to the subject.

We begin with a brief review of solutions to second-order ordinary differential equations

We then consider single and multiple degree of freedom systems We conclude with abrief discussion of nonlinear vibrations

13.2 Solutions of Second-Order Differential Equations

Vibration phenomena are often modeled by second-order ordinary differential equations.Solutions of these equations provide a representation of the movement of vibrating sys-tems; therefore, to begin our study, it is helpful to review the solution procedures of second-order ordinary differential equations The reader is encouraged to also independentlyreview these procedures References 13.1 to 13.7 provide a sampling of the many textsavailable on the subject

We will consider first the so-called linear oscillator equation:

(13.2.1)where, as before, the overdot represents differentiation with respect to time, and ω is aconstant

In Eq (13.2.1) the time t is the independent variable and x is the dependent variable to

be determined It is readily verified that the solution of Eq (13.2.1) may be expressed inthe form:

440 Dynamics of Mechanical Systems

where A and B are constants that may be evaluated by auxiliary conditions (initial tions or boundary conditions) (We can verify that the expression of Eq (13.2.2) is indeed asolution of Eq (13.2.1) by direct substitution It is in fact a general solution, as cosωt andsinωt are independent functions and there are two arbitrary constants, A and B.)

condi-Through use of trigonometric identities, we can express Eq (13.2.2) in the form:

(13.2.3)where  and φ are constants To develop this, recall the identity:

(13.2.4)Then, by thus expanding the expression of Eq (13.2.3) we have:

of  and – Also, the period T of the oscillation is determined by:

cos(α β+ )≡cos cosα β−sin sinα β

π/2

π

Introduction to Vibrations 441

As noted earlier, the constants in Eqs (13.2.2) and (13.2.3) are to be evaluated by auxiliaryconditions These auxiliary conditions are generally initial conditions or boundary condi-tions Initial conditions (where t = 0) might be expressed as:

(13.2.9)Then, by requiring the solution of Eq (13.2.2) to meet these conditions, we have:

(13.2.10)and thus,

(13.2.11)Boundary conditions might be expressed as:

(13.2.12)

Then, by requiring the solution of Eq (13.2.2) to meet these conditions, we see that theconstants A and B must satisfy:

(13.2.13)The second expression is satisfied by either:

442 Dynamics of Mechanical Systems

By comparing the solution procedures between the initial and boundary value problems,

we see that the solution of the initial value problem is simpler and more direct than that

of the boundary value problem Therefore, for simplicity, in the sequel we will consider

primarily initial value problems

Consider next the damped linear oscillator equation:

(13.2.18)where m, c, and k are constants This is the classical second-order homogeneous linear

ordinary differential equation with constant coefficients From References 13.9 to 13.13,

we see that the solution depends upon the relative magnitudes of m, c, and k If the product

4km is less than c2, we can write the solution in the form:

(13.2.19)where µ and ω are defined as:

(13.2.20)

If, in Eq (13.2.18), the product c2 exceeds 4km, the solution takes the form:

(13.2.21)where µ and ν are defined as:

(13.2.22)

Finally, if in Eq (13.2.18), the product 4km is exactly equal to c2, the solution takes the

form:

(13.2.23)where µ is:

(13.2.24)Next, consider the forced linear oscillator described by the equation (with 4km > c2):

(13.2.25)where f(t) is the forcing function The forcing function typically has the form:

2

4

2 2

2

4

2 2

Introduction to Vibrations 443

Equation (13.2.25) then becomes:

(13.2.27)From References 13.9 to 13.13, we see that the general solution may be expressed as:

(13.2.28)

where x h is the general solution of the homogeneous equation (right side equal to zero)

as in Eqs (13.2.18) and (13.2.19), and where x p is any solution of the nonhomogeneous

equation (right side equal to Fcospt, as in Eq (13.2.27)) x p is commonly called the particular

solution From Eq (13.2.19) we see that x h is:

(13.2.29)

where, as before, µ and ω are defined by Eq (13.2.20) Also, from the references, we see

that x p may be expressed as:

(13.2.30)where ∆ is defined as:

(13.2.31)

(The validity of Eq (13.2.26) may be verified by direct substitution into Eq (13.2.23).)

Finally, if c is zero in Eq (13.2.27), we have the forced undamped linear oscillator

equation:

(13.2.32)From Eqs (13.2.28), (13.2.29), and (13.2.30), we see that the solution is:

(13.2.33)

where x h and x p are:

(13.2.34)and

(13.2.35)where from Eq (13.2.20) ω is defined as:

13.3 The Undamped Linear Oscillator

Consider the undamped linear oscillator consisting of the mass–spring system, which we

considered in Chapter 11, Section 11.5, and as shown in Figure 13.3.1, where m is the mass

of a block B sliding on a smooth (frictionless) horizontal surface, k is the modulus of a linear supporting spring, and x measures the displacement of B away from its equilibrium configuration The system is said to be undamped because B moves on a frictionless surface

and the total energy of the mass–spring system is unchanged during the motion Usingany of the principles of dynamics, we find the equation of motion to be:

(13.3.1)or

(13.3.2)where:

(13.3.3)From Eq (13.2.2), we see that the solution of Eq (13.3.2) is:

(13.3.4)

where, as we noted, A and B are constants to be evaluated from auxiliary conditions such

as initial conditions for the mass–spring system For example, suppose that at time t = 0

the displacement and displacement rate are:

(13.3.5)Then, from Eq (13.3.4), we have:

(13.3.6)Hence, the solution becomes:

where f is the frequency (or natural frequency) and T is the period.

Consider next the simple pendulum as in Figure 13.3.2 From Eq (12.2.9), we see thatthe equation of motion is:

(13.3.11)

where as before, θ measures the displacement away from equilibrium, g is the gravity

constant, and  is the pendulum length Observe that if θ is small, as is the case with mostpendulums, we can approximate sinθ by the first few terms of a Taylor series expansion

of sinθ about the equilibrium position θ = 0 That is,

(13.3.12)Hence, for small θ, we have:

(13.3.13)The governing equation Eq (13.3.11), then becomes:

Equation (13.3.14) is identical in form to Eq (13.3.2); therefore, the solution will havethe form of Eq (13.3.7) That is,

(13.3.15)where θ0 and 0 are the initial values of θ and , respectively, and where ω is:

(This is the reason why pendulums have been used extensively in clocks and timing devices.)

13.4 Forced Vibration of an Undamped Oscillator

Consider again the undamped mass–spring system of the foregoing section and as

depicted again in Figure 13.4.1 This time, let the mass B be subjected to a time-varying force F(t) as shown Then, it is readily seen by using any of the principles of dynamics

discussed earlier, that the governing equation of motion for this system is:

Introduction to Vibrations 447

where F0 and p are constants The governing equation then becomes:

(13.4.3)From Eqs (13.2.32) through (13.2.35) we see that the general solution of Eq (13.4.3) is:

(13.4.4)

where as before A and B are constants to be determined from auxiliary conditions (such

as initial conditions) and where ω is defined as:

(13.4.5)

Consider the last term, [F0/(k – mp2)]cospt of Eq (13.4.4) Observe that if p2 has values

nearly equal to k/m (that is, if p is nearly equal to ω) the denominator becomes very small,

producing large-amplitude oscillation Indeed, if p is equal to ω, the oscillation amplitude

becomes unbounded This means that by stimulating the mass B with a periodic force

having a frequency nearly equal to ω (that is, ), the amplitude of the resulting oscillationbecomes unbounded

The quantity is called the natural frequency of the system When the frequency of

the loading function is equal to the natural frequency, giving rise to a large-amplitude

response, we have the phenomenon commonly referred to as resonance.

13.5 Damped Linear Oscillator

Consider next the damped linear oscillator as depicted in Figure 13.5.1 This is the same

system we considered in the previous sections, but here the movement of the mass B is

restricted by a “damper” in the form of a dashpot For simplicity of illustration, we will

assume viscous damping, where the force exerted by the dashpot on B is proportional to the speed of B and directed opposite to the motion of B with c being the constant of

proportionality That is, the damping force FD on B is:

By considering a free-body diagram of B and by using the principles of dynamics (for

example, Newton’s laws or d’Alembert’s principle), we readily see that the governingequation of motion is:

(13.5.2)

where as before, m is the mass of B and k is the linear spring modulus.

From Eqs (13.2.18), (13.2.19), and (13.2.20), we see that the solution of Eq (13.5.2) may

be written in the form:

(13.5.3)where µ and ω are defined as:

(13.5.4)

and where, as before, A and B are constants to be evaluated from auxiliary conditions,

such as initial conditions, on the system

Observe in Eq (13.5.3) that if the damping constant c is small we have an oscillating

system where the amplitude of the oscillation slowly decreases, as in Figure 13.5.2 This

phenomenon is called underdamped vibration.

Next, observe in Eqs (13.5.3) and (13.5.4) that if the damping constant c is such that ω

is zero, there will be no oscillation That is, if

(13.5.5)then

(13.5.6)and

2

4

2 2

1 2 /

c= 2 km

ω = 0

x=e−µt(A Bt+ )

Introduction to Vibrations 449

where µ is:

(13.5.8)

In this case, the block moves to its static equilibrium position without oscillation This

phenomenon is called critical damping.

Finally, suppose the damping constant is larger than critical damping, that is, largerthan 2 Then, from Eq (13.5.4), we see that ω is imaginary and that the solution for

the displacement x of B becomes:

(13.5.9)where µ and ν are:

(13.5.10)

Observe that µ is always larger than ν and we have a relatively rapidly decaying motion

of B This phenomenon is called overdamping.

13.6 Forced Vibration of a Damped Linear Oscillator

Consider next the forced vibration of a damped linear oscillator as depicted in Figure13.6.1 From the principles of dynamics, we readily find the governing equation of motion

2

4

2 2

Equation (13.6.1) then takes the form:

(13.6.6)where µ, ω, and ∆ are defined by:

(13.6.7)

where we assume that c2 < 4km As before, the constants A and B in Eq (13.6.5) are to be

evaluated by auxiliary (initial) conditions on the system

Observe that the last term of Eq (13.6.6) can become quite large if the damping coefficient

c is small and if the frequency p of the forcing function is nearly equal to the natural

frequency of the undamped system Note, however, that unlike the undampedsystem, the presence of damping assures that the amplitude of the oscillation remainsfinite Thus, we see that damping (or effects of friction and viscosity) can have a beneficialeffect in preventing harmful or unbounded vibration of a mechanical system

The phenomenon of damping in physical systems, however, is generally more complexthan our relatively simple model of Eq (13.5.1) Indeed, damping is generally a nonlinearphenomenon that varies from system to system and is generally not well understood.Theoretical and experimental research on damping is currently a major interest of vibrationanalysts

13.7 Systems with Several Degrees of Freedom

We consider next mechanical systems where more than one body can oscillate An example

of such a system might be a double mass–spring system as in Figure 13.7.1 This system

has two degrees of freedom as represented by the displacements x1 and x2 of the masses.Accordingly, we expect to obtain two governing differential equations that must be solved

2

4

2 2

Introduction to Vibrations 451

simultaneously Similarly, systems with three or more degrees of freedom will have three

or more governing differential equations to be solved simultaneously

To illustrate a procedure for studying such systems, consider again the system of connected smooth particles (or balls) in the rotating tube as in Figure 13.7.2 As before,

spring-let each particle have mass m and spring-let the connecting springs be linear with natural length

 and modulus k.

To simplify our analysis, let θ be fixed at 90° so that the particles move in a fixed

horizontal tube with their position defined by the coordinates x1, x2, and x3 as in Figure13.7.3 From Eqs (12.2.27), (12.2.28), and (12.2.29), we see that, with θ fixed at 90°, theequations of motion may be written as:

(13.7.1)(13.7.2)(13.7.3)Equations (13.7.1), (13.7.2), and (13.7.3) may be written in the matrix form:

To solve Eq (13.7.4), let x have the matrix form:

(13.7.7)where ω is a frequency to be determined and A is the array:

(13.7.11)That is,

As with the eigenvalue problem we encountered in Sections 7.7, 7.8, and 7.9 in studying

inertia, we have a system of simultaneous linear algebra equations for the amplitudes A1,

A2, and A3 The system is homogeneous in that the right sides are zero Thus, we have anontrivial (or nonzero) solution only if the determinant of the coefficients is zero Hence,

Observe that instead of obtaining one solution we have three solutions That is, there

are three positive frequencies (three natural frequencies) that make the determinant of Eq.

(13.7.16) equal to zero and thus allow a nonzero solution of Eqs (13.7.13), (13.7.14), and(13.7.15) to occur This in turn means that we can expect to find three sets of amplitudessolving Eqs (13.7.13), (13.7.14), and (13.7.15), one set for each frequency ωi (i = 1, 2, 3).

To find these amplitude solutions, we can select one of the ωi (say, the smallest of the

ωi ), substitute it into Eqs (13.7.13) to (13.7.15), and thus solve for the amplitudes A i (i =

1, 2, 3) We can then repeat the process for the other two values of ωi Notice, however,that although there are three equations for the three amplitudes, the equations are notindependent in view of Eq (13.7.16) That is, there are at most two independent equations,thus we need another equation to obtain a unique solution for the amplitudes Such anequation can be obtained by arbitrarily specifying the magnitude of one or more of theamplitudes For example, we could “normalize” the amplitudes such that

(13.7.19)

Then, Eqs (13.7.13), (13.7.14), (13.7.15), and (13.7.19) are equivalent to an independent set

of three equations, enabling us to determine unique values of the amplitudes

To this end, let us select the smallest of the ωi (ω1) and substitute it (that is, {(2 – )

(k/m)}1/2) into Eqs (13.7.13), (13.7.14), and (13.7.15) This produces the equations:

(13.7.20)

(13.7.21)

(13.7.22)

Observe that these equations are dependent (If we multiply the first and third equations

by and add them we obtain a multiple of the second equation.) Thus, by selecting two(say, the first two) of these equations and by combining them with Eq (13.7.19), we obtainupon solving the expressions:

3 2

= −( ) ( )k m , = ( )k m , = +( ) ( )k m

A12 A A

2 2 3

A1=1 2, A2= 2 2, A3=1 2

Next, let ω be ω2 (that is, ) By substituting into Eqs (13.7.13), (13.7.14), and(13.7.15), we obtain the equations:

(13.7.24)(13.7.25)(13.7.26)These equations are also dependent (the first and third are the same) By using the first

two of these together with Eq (13.7.19) and solving for A1, A2, and A3, we have:

Introduction to Vibrations 455

13.8 Analysis and Discussion of Three-Particle Movement:

Modes of Vibration

In reviewing the foregoing analysis of the three-particle system we see a striking similarity

to the analysis for the eigenvalue problem of Sections 7.7, 7.8, and 7.9 Indeed, a comparison

of Eqs (13.7.13), (13.7.14) and (13.7.15) with Eq (7.7.10), shows that they are in essencethe same problem This means that analyses similar to those of Sections 7.7, 7.8, and 7.9(such as determination of eigenvalues, eigenvectors, orthogonality, etc.) could also beconducted for the three-particle vibration problem In our relatively brief introduction tovibrations, however, it is not our intention to develop such detail Instead, we plan tosimply introduce the concept of vibration modes (analogous to eigenvectors)

To this end, consider again the governing differential equations for the spring-supportedparticles (see Eqs (13.7.1), (13.7.2), and (13.7.3)):

(13.8.1)(13.8.2)(13.8.3)

Recall that we found not one but three nontrivial solutions to these equations Each solution

had its own frequency, which means that the system can vibrate in three ways, or in three

“modes,” as depicted in Figures 13.7.4, 13.7.5, and 13.7.6 These are called the natural modes

of vibration of the system.

To discuss this further, consider the amplitudes of these vibration modes as in Eqs.(13.7.23), (13.7.27), and (13.7.31) and as listed in Table 13.8.1 In view of the amplituderatios, let new variables ξ1, ξ2, and ξ3 be introduced as:

(13.8.4)

(13.8.5)

(13.8.6)Then, by differentiating, we have:

(13.8.7)

(13.8.8)

(13.8.9)Consider first 1 By substituting from Eqs (13.8.1), (13.8.2), and (13.8.3), we have:

TABLE 13.8.1

Modes of Vibration of Spring-Supported Particles

Mode Frequency Normalized Amplitudes