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4.4 Torsional Vibration of Shafts

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Torsional vibrations are oscillating angular motions of a device about some axis of rotation. Examples are

vibration in shafts, rotors, vanes, and propellers. The governing PDE of the torsional vibration of a shaft

is quite similar to that we previously encountered of the transverse vibration of a cable in tension and the

longitudinal vibration of a rod. However, in the present case, the vibrations are rotating (angular)

motions with resulting shear strains, shear stresses, and torques in the torsional member. Furthermore,

the parameters of the equation of motion will take different meanings. When bending and torsional

motions occur simultaneously, there can be some interaction, thereby making the analysis more difficult.

Here, we neglect such interactions by assuming that only the torsional effects are present or that the

motions are quite small.

Since the form of the torsional vibration equation is similar to forms we have studied before, the same

procedures of analysis may be employed and, in particular, the concepts of modal analysis will be similar.

However, the torsional parameters will be rather complex for members with noncircular cross sections.

Nevertheless, vast majority of torsional devices have circular cross sections.

4.4.1 Shaft with Circular Cross Section

Here, we will formulate the problem of the torsional vibration of a shaft having a circular cross section.

The general case of a nonuniform cross section along the shaft is considered, but the usual assumptions

such as homogeneous, isotropic, and elastic material are made.

First, we will obtain a relationship between torque ðTÞ and angular deformation or twist ðuÞ for a

circular shaft. Consider a small element of length dx along the shaft axis and the cylindrical surface at a

general radius r (in the interior of the shaft segment), as shown in Figure 4.6(a). During vibration, this

element will deform (twist) through a small angle du:

A point on the circumference will deform through r du as a result, and a longitudinal line on the

cylindrical surface will deform through angle g; as shown in Figure 4.6(a). From the strength of materials

and elasticity theory of solid mechanics, we know that g is the shear strain. Hence,

Shear strain g ¼

r du

γ r dθ

(a) dx (b)

T T+dT

r+dr Shear stress

FIGURE 4.6 (a) Small element of a circular shaft in torsion; (b) shear stresses in a small annular cross section

carrying torque.

Distributed-Parameter Systems 4-19

However, allowing for the fact that the angular shift u is a function of t as well as x in the general case of

dynamics, we use partial derivatives and write

g ¼ r

›u

›x ð4:87Þ

The corresponding shear stress at the deformed point at radius r is

t ¼ Gg ¼ Gr

›u

›x ð4:88Þ

where G ¼ shear modulus.

This shear stress acts tangentially. Consider a small annular cross section of width dr at radius r of

the shaft, as shown in Figure 4.6(b). By symmetry, the shear stress will be the same throughout this

region and will form a torque of rt £ 2pr dr ¼ 2pr2t dr: Hence, the overall torque at the shaft cross

section is

T ¼

2pr2t dr

which, in view of Equation 4.88, is written as

T ¼ G

›u

›x

2pr3 dr ð4:89Þ

It is clear that the integral term is the polar moment of area of the shaft cross section:

J ¼

2pr3 dr ð4:90Þ

In particular, for a solid shaft of radius r

J ¼

pr4

2 ð4:91Þ

and, for a hollow shaft of inner radius r1 and the outer radius r2

J ¼

2 ðr4

2 2 r4

1Þ ð4:92Þ

So, we write Equation 4.89 as

T ¼ GJðxÞ

›u

›x ð4:93Þ

The combined parameter GJ is termed the torsional rigidity of the shaft. We have emphasized that the

shaft may be nonuniform and hence J is a function of x: Consider a uniform shaft segment of length l;

with associated overall angular deformation u: Equation 4.93 can be written as

Torsional stiffness K ¼

u ¼

l ð4:94Þ

Note: For a shaft with noncircular cross section, replace J by Jt in this equation. It follows that, the

larger the torsional rigidity GJ; the higher the torsional stiffness K; as expected. Furthermore, longer

members have a lower torsional stiffness (and smaller natural frequencies).

Now, we apply Newton’s Second Law for rotatory motion of the small element dx; shown in

Figure 4.6(a). The polar moment of inertia of the element is

r2 dm ¼

r2r dx dA ¼ r dx

r2 dA ¼ rJ

dx; where J is the polar moment of area, as discussed before. Also, suppose that a distributed external

torque of tðx; tÞ per unit length is applied along the shaft. Hence, the equation of motion is

rJ dx

›2u

›t2 ¼ T þ dT 2 T þ tðx; tÞdx ¼

›T

›x

dx þ tðx; tÞdx

4-20 Vibration and Shock Handbook

Substitute Equation 4.93 and cancel dx to obtain the equation of torsional vibration of a circular

shaft as

›2uðx; tÞ

›t2 ¼

›

›x

GJðxÞ

›uðx; tÞ

›x þ tðx; tÞ ð4:95Þ

For the case of a uniform shaft (constant J) in free vibration ðtðx; tÞ ¼ 0Þ; we have

›2uðx; tÞ

›t2 ¼ c2 ›2uðx; tÞ

›x2 ð4:96Þ

with

c ¼

ffiffiffiffi

ð4:97Þ

Note that Equation 4.96 is quite similar to that for transverse vibration of a cable in tension and the

longitudinal vibration of a rod. Hence, the same concepts and procedures of analysis may be used. In

particular, two boundaries conditions will be needed in the solution; for example

Fixed end at x ¼ x0: uðx0; tÞ ¼ 0 ð4:98Þ

Free end at x ¼ x0:

›uðx0; tÞ

›x ¼ 0 ð4:99Þ

The orthogonality property of mode shapes QiðxÞ:

ðl

QiðxÞQjðxÞdx ¼

0 for i – j

lj for i ¼ j

(

ð4:100Þ

4.4.2 Torsional Vibration of Noncircular Shafts

Unlike the longitudinal and transverse vibrations of rods and beams, when considering the torsional

vibration of shafts, the equation of motion for circular shafts (Equation 4.95 and Equation 4.96) cannot

be used for shafts with noncircular cross sections. The reason is that the shear stress distributions in the

two cases can be quite different, and Equation 4.88 does not hold for noncircular sections. Hence, the

parameter J in the torque – deflection relations (e.g., Equation 4.93 and Equation 4.94) is not the polar

moment of area in the case of noncircular sections. In the noncircular case, we write

T ¼ GJt

›u

›x ð4:101Þ

where Jt ¼ torsional parameter.

The Saint-Venant theory of torsion and the related membrane analogy, developed by Prandtl, have

provided equations for Jt in special cases. For example, for a thin hollow section

Jt ¼

4tA2s

p ð4:102Þ

where

As ¼ enclosed (contained) area of the hollow section

p ¼ perimeter of the section

t ¼ wall thickness of the section

For a thin, solid section, we have

Jt ¼

t3a

3 ð4:103Þ

Distributed-Parameter Systems 4-21

where

a ¼ length of the narrow section

t ¼ thickness of the narrow section

Torsional parameters for some useful sections are given in Table 4.2.

TABLE 4.2 Torsional Parameters for Several Sections

Section Shape Torsional Parameter Jt

Solid circular

Hollow circular

2 ðr4

2 2 r4

1 Þ

Thin closed

4tA2s

Thin open t

t3a

Solid square

0:1406a4

Hollow square

a1 a1

0:1406ða42

2 a41

4-22 Vibration and Shock Handbook

Example 4.4

Consider a thin, rectangular hollow section of thickness t; height a; and width a=2; as shown in

Figure 4.7(a). Suppose that the section is opened by making a small slit as in Figure 4.7(b). Study the

affect on the torsional parameter Jt and torsional stiffness K of the member due to the opening.

Solution

(a) Closed section:

The contained area of the section is As ¼ a2=2

The perimeter of the section p ¼ 3a

Using Equation 4.102, the torsional parameter is

Jtc ¼

ta3

(b) Open section:

The solid length of the section ¼ 3a

Using Equation 4.103, the torsional parameter is

Jt0 ¼ at3

The ratio of the torsional parameters is

Jt0

Jtc ¼

3t2

For members of equal length, torsional stiffness will also be in the same ratio as is given by this

expression. Since t is small compared with a; there will be a significant drop in torsional stiffness due to

the opening (cutout).

Example 4.5

An innovative automated transit system uses an elevated guideway with cars whose suspension is

attached to (and slides on) the side of the guideway. Owing to this eccentric loading on the guideway,

a/2

(a) (b)

FIGURE 4.7 (a) A thin closed section; (b) a thin open section.

Distributed-Parameter Systems 4-23

there is a significant component of torsional dynamics in addition to bending. Assume that the torque Tj

acting on the guideway due to the jth suspension of the vehicle to be constant, acting at a point xj as

measured from one support pier, and moving at speed vj: A schematic representation is given in

Figure 4.8. The guideway span shown has a length L and a cross section which is a thin-walled rectangular

box of height a; width b; and thickness t: The ends of the guideway span are restrained for angular motion

(i.e., fixed):

(a) Formulate and analyze the torsional (angular) motion of the guideway.

(b) For a single point vehicle entering a guideway that is at rest, what is the resulting dynamic

response of the guideway? What is the critical speed that should be avoided?

l ¼ 60 ft ð18:3 mÞ

abt ¼ 5 ft £ 2:2 ft £

ft ð1:52 m £ 0:67 m £ 0:15 mÞ

r ¼ 4:66 slugs=ft3 ð2:4 £ 103 kg=m3Þ

G ¼ 1:55 £ 106 lb=in:2 ð1:07 £ 1010 N=m2Þ

and vehicle speed

v ¼ 60 mi=h ð26:8 m=secÞ

Compute the crossing frequency ratio given by

vc ¼

Rate of span crossing

Fundamental natural frequency of guideway

and discuss its implications.

Solution

(a) For a uniform guideway with distributed torque load tðx; tÞ and a noncircular cross section having

torsional parameter Jt ; the governing equation is

›2uðx; tÞ

›t2 ¼ GJt

›2u

›x2 þ tðx; tÞ ð4:104Þ

As usual, the mode shapes are obtained by solving

d2Q

dx2 þ l2Q ¼ 0 ð4:105Þ

and the corresponding natural frequencies are given by

vi ¼ li

ffiffiffiffiffiffiffiffi

GJ=rJt

ð4:106Þ

Tj vj

Support

Pier

Support

Pier

Guideway

Car

FIGURE 4.8 A torsional-guideway transit system.

4-24 Vibration and Shock Handbook

The general solution of Equation 4.105 is

QðxÞ ¼ A1 sin lx þ A2 cos lx

where Ai ði ¼ 1; 2Þ are the constants of integration. The torsional BCs corresponding to the fixed

ends (no twist) are

Qð0Þ ¼ QðlÞ ¼ 0

where l ¼ guideway span length. For a nontrivial solution, we need

A2 ¼ 0

and

li ¼

i ¼ 1; 2; … ð4:107Þ

The solution corresponds to an infinite set of eigenfunctions QiðxÞ satisfying Equation 4.105. Each

of these represents a natural mode in which the beam can undergo free torsional vibrations. The

actual motion consists of a linear combination of the normal modes depending on beam initial

conditions and the forcing term tðx; tÞ: The integration constant A1 may be incorporated

(partially) into the generalized coordinate q; which is still unknown and is determined through

initial conditions. Here, we use the normalized eigenfunctions

QiðxÞ ¼

ffiffi

2 p sin

ipx

i ¼ 1; 2; … ð4:108Þ

The orthogonality condition given by

ðl

QiQj dx ¼

0 for i – j

1 for i ¼ j

(

ð4:109Þ

is satisfied. In view of the relations (Equation 4.106 and Equation 4.107) the natural frequencies

corresponding to different eigenfunctions (natural mode shapes) are

vi ¼

ffiffiffiffiffiffiffiffi

GJ=rJt

i ¼ 1; 2; … ð4:110Þ

For n number of vehicle’s suspensions located on the analyzed span

tðx; tÞ ¼

j¼1

Tjdðx 2 x0j 2 vjtÞ ð4:111Þ

where

Tj ¼ torque exerted on guideway by the jth suspension

vj ¼ speed of the jth suspension

x0j ¼ initial position (at t ¼ 0) along the guideway of the jth suspension

dð·Þ ¼ Dirac delta function

The forced motion can be represented in terms of the normalized eigenfunction as

uðx; tÞ ¼

i¼1

qiðtÞQiðxÞ ð4:112Þ

where qiðtÞ ¼ the generalized coordinate for forced motion in the ith mode. On substituting the

relations (Equation 4.111 and Equation 4.112) into Equation 4.104, and integrating the result over

the span length, after multiplying by a general eigenfunction while making use of the

Distributed-Parameter Systems 4-25

orthogonality relation (Equation 4.109), one obtains

d2qi

dt2 þ v2i

qi ¼

rJt l

j¼1

TjQiðx0j þ vjtÞ i ¼ 1; 2; … ð4:113Þ

(b) For a single suspension entering the guideway at t ¼ 0; with the guideway initially at

rest ðqið0Þ ¼ q_ið0Þ ¼ 0Þ; we have n ¼ 1 and x01 ¼ 0: Then, the complete solution of

Equation 4.113 is

qiðtÞ ¼

ffiffi

2 p lT sin

ipvt

2 2vc sin vit

􀀒 􀀓

GJt p2i2ð1 2 v2

C Þ

i ¼ 1; 2; … ð4:114Þ

where the crossing-frequency ratio vC is given by

vC ¼

lv1 ð4:115Þ

Note from Equation 4.114 that the critical speed corresponds to vC ¼ 1 and should be

avoided. In typical transit systems, vC is considerably less than 1.

Jt ¼ 3:485 £ 105 in:4 ð1:451 £ 107 cm4Þ

J ¼ 4:574 £ 105 in:4 ð2:736 £ 107 cm4Þ

v1 ¼ 263:8 rad=sec ¼ 42:0 Hz

Note: We used the expression for a thin, hollow section, given in Table 4.2, in computing Jt : Finally,

the crossing-frequency ratio is computed to be vC ¼ 0:017, which is much less than 1.0, as expected.

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