Please cite this article as:
Stanisław Kukla, Izabela Zamojska, Free longitudinal vibrations of multi-step non-uniform rods, Scientific Research of
the Institute of Mathematics and Computer Science, 2004, Volume 3, Issue 1, pages 73-80.
The website: http://www.amcm.pcz.pl/
Scientific Research of the Institute of Mathematics and Computer Science
FREE LONGITUDINAL VIBRATIONS
OF MULTI-STEP NON-UNIFORM RODS
Stanisław Kukla, Izabela Zamojska
Institute of Mathematics and Computer Science, Czestochowa University of Technology
Abstract. The paper presents the solution of the free longitudinal vibration problem for
a multi-step rod with varying cross-section areas. The problem considered takes into
account vibrations of rods with additional discrete spring-mass elements attached at their
ends. The problem solution is obtained with the application of the Green’s function properties and the power series method. The presented numerical example shows the influence
of the selected parameter characterizing the discrete element on free vibration frequencies
of the system.
1. Introduction
The vibration analysis of non-uniform rods is the subject of papers [1-6]. The
solution of the vibration problem in a closed form can be found in particular cases
of the cross-sectional area and the mass per unit length of the rod. It is often assumed that the mass is constant and the cross-sectional area varying with respect to
the space variable. In this case the solution of the free vibration problem is well
known for katenoidal, sinusoidal and exponentional rods [1]. For the rod with
a cross-sectional area A(ξ) = (αξ + β)c, where c is a real number, a solution has
been presented by Kumar [2] and Li [3]. The free vibration problem of a system of
non-uniform rods coupled by springs was the subject of paper [5]. Such problems
can be solved with the application of the Green’s function method.
The Green’s function method was used in paper [7] to solve the vibration problem of a system of two uniform rods coupled by translational springs. The frequency equation and mode shapes are expressed by Green’s functions corresponding to rods which are components of the considered combined system. The application of the method is possible only if the right Green’s functions are known.
A solution of the free vibration problem for non-uniform rods carrying an arbitrary
number of discrete elements was presented in reference [6].
In this paper, the Green’s function method is used to solve the free vibration
problem of a rod consisting of any number of non-uniform segments. The necessary Green’s functions have been derived with the use of a power series method.
An example of numerical calculations of free longitudinal vibration frequencies
of a multi-step rod is given.
74
S. Kukla, I. Zamojska
2. Formulation and solution of the problem
Consider a rod consisting of n non-uniform segments as shown in Figure 1.
Assume that a discrete spring-mass element is attached at point x = L. Free vibration of the rod is governed by the following differential equations:
L1 u1 ( x, t )  = s1 (t )δ ( x − L1 ) for x ∈ [0,L1]
(1)
Li ui ( x, t )  = − si −1 (t )δ ( x − Li −1 ) + si (t )δ ( x − Li ) x ∈ [Li−1,Li], i = 2,..., n−1
L n un ( x, t )  = − sn −1 (t )δ ( x − Ln −1 ) + p ( x, t ) for x∈[Ln−1,L]
(2)
(3)
∂
∂
∂2
[ E Ai ( x ) ] − ρ Ai ( x ) 2 for i = 1,2,…, n, Ai ( x ) is the area of
∂x
∂x
∂t
the cross-section at point x of the rod, E is the modulus of elasticity, ρ is
the mass density of the rod material, δ( ) denotes the Dirac delta function,
 ∂ 2un ( x, t )

+ kun ( x, t ) δ ( x − L ) , m and k are the discrete mass and the
p( x, t ) =  m
2
∂t


stiffness coefficient of the spring attached at point x = L, respectively.
The functions ui satisfy homogeneous boundary conditions which may be symbolically written in the following form:
where Li =
m
k
x
L1
L2
L3
Ln = L
Fig. 1. A sketch of a multi-step non-uniform rod with a spring-mass
discrete element
Free Longitudinal Vibrations of Multi-Step Non-Uniform Rods
B0 [u1 ]
x=0
75
(4)
= 0 B1[un ] x = L = 0
where B0 and B are linear, spatial differential operators and L denotes the length
of the rod. Moreover, at points Li the following continuity conditions are satisfied:
1
i = 1,..., n−1
ui ( Li , t ) = ui +1 ( Li , t ),
(5)
Assuming that the natural frequencies of the rod are harmonic, i.e. substituting
the functions:
ui ( x, t ) = U i ( x)e jω t , p( x, t ) = P( x)e jω t , si (t ) = S i e jω t
(6)
into equations (1)-(3), one obtains:
L% 1 [U1 ] = S1δ (ξ − l1 ) for ξ ∈ [0, l1 ]
(7)
L% i [U i ] = − µi −1Si −1δ (ξ ) + Siδ (ξ − li ) for ξ ∈ [0, li ] , i = 2,..., n−1
(8)
L% n [U n ] = − µ n −1Sn −1δ (ξ ) + P(ξ ) for ξ ∈ [0, ln ]
(9)
where: j 2 = −1, Ω2 = ρω 2 E , Λ 2 = ρω 2 E , ω = k m , Si = S i EAi (0), ω are
the natural frequencies of the rod, ξ = x − Li −1 , li = Li−Li−1, η = m ρ An (0),
Ai (ξ ) = Ai ( x ) Ai ( 0 ) ,
P (ξ ) = η (Λ 2 − Ω2 )U n (ξ )δ (ξ − ln ),
µi −1 = Ai −1 (0) Ai (0),
d 
d 
2
L% i =
 Ai (ξ )  + Ωi Ai (ξ ) for ξ ∈ [0, li ] , i = 1,...n.
dξ 
dξ 
The boundary and continuity conditions (4), (5) may be written in the following
form:
(10)
B0 [U1 ] ξ = 0 = 0, B1[U n ] ξ = l = 0
n
U i (li ) = U i +1 (0), i = 1,..., n−1
(11)
The solution of problems (7)-(11) is obtained with the use of the properties of
Green’s functions Gi. If the Green’s functions are known, then the following relationships are obtained on the basis of equations (7)-(9):
U1 (ξ ) = S1G1 (ξ , l1 )
(12)
U i (ξ ) = − µ i −1Si −1Gi (ξ , 0) + Si Gi (ξ , li ) for i = 2,...n−1
(13)
U (ξ ) = − µ S G (ξ , 0) + η ( Λ − Ω )U (l )G (ξ , l )
(14)
2
n
n − 1
n − 1
n
2
n
n
n
n
76
S. Kukla, I. Zamojska
A system of equations is obtained by substituting functions Ui(ξ) into continuity
conditions (11):
S1[G1 (l1 , l1 ) + µ1G2 (0, 0)] − S 2G2 (0, l2 ) = 0
(15)
− µ i −1Si −1Gi (li , 0) + Si [Gi (li , li ) + µ i Gi +1 (0, 0)] − Si +1Gi +1 (0, li +1 ) = 0, i = 2,3,...n−2 (16)
An additional equation is obtained by setting in equation (14) ξ = ln:
µ n−1S n−1Gn (ln , 0) + [1 + η (Ω2 − Λ 2 )Gn (ln , ln )]U n (ln ) = 0
(17)
The set of equations (15)-(17) may be written in the matrix form
A(ω) S = 0
(18)
where S = [S1, S2,…, Sn−1, Un(ln)]T, A = [aij ] is n x n dimensional matrix with
aii −1 = −µ i −1Gi (li , 0) for i = 2,3,...,n; aii = Gi (li , li ) + µiGi +1 (0,0) for i = 1,2,...,n − 1;
aii +1 = −Gi +1 (0, li +1 ) for i = 1,2,...,n − 2; an −1 n = η (Ω2 − Λ 2 )Gn (0, ln ); an n = 1 +
+ η (Ω2 − Λ 2 )Gn (ln , ln ) and the remaining coefficients aij are equal to zero.
For a non-trivial solution of the problem, the determinant of the coefficient matrix is set equal to zero, yielding the equation
det A(ω) = 0
(19)
Equation (19) (the equation of natural frequencies of the rod, with unknown eigenfrequencies ω) is then solved numerically.
3. The use of the power series method in deriving
the Green’s functions
The solution Ui(ξ) of the eigenproblem (7)-(11) was obtained with the use of
known Green’s functions. The Green’s functions Gi satisfy the equation:
L% i Gi (ξ , ζ )  = δ (ξ − ζ ) , i = 1,2,…,n
(20)
Functions G1, Gn satisfied the boundary conditions (10) and Gi - the conditions
corresponding to the free ends of the rod, i.e.:
Gi ,ξ
ξ =0
= 0, for i = 2,3,...,n, and Gi ,ξ
The function Gi may be written in the form
ξ = li
= 0, for i = 1,2,...,n−1
(21)
Free Longitudinal Vibrations of Multi-Step Non-Uniform Rods
Gi (ξ , ζ ) = Gi0 (ξ , ζ ) + Gi1 (ξ , ζ ) H (ξ − ζ )
77
(22)
where Gi0 (ξ , ζ ) is a general solution of the homogeneous equation:
L% i Gi (ξ , ζ )  = 0, i = 1,2,…,n
(23)
and Gi1 (ξ , ζ ) H (ξ − ζ ) is a particular solution of equation (20). It may be proved
that the functions Gi1 (ξ , ζ ) = Gi1 (ξ − ζ ) are solutions of equation (23) which satisfy
the following conditions:
Gi1
ξ =ζ
dGi1
dξ
= 0,
=
ξ =ζ
1
Ai (ζ )
(24)
The general solution V (ξ ) of the differential equation L% V (ξ )  = 0 will be
determined assuming that function Ai (ξ ) is expressed as (the index i is omitted)
∞
A(ξ ) =
ar
∑ r! ξ
r
(25)
r =0
The function V (ξ ) is searched with the use of the power series method
∞
V (ξ ) =
υr
∑ r! ξ
r
(26)
r =0
Substituting functions A(ξ) and V(ξ) into equation (23), one obtains:
r
 r + 1
r
2
a
υ
+
Ω

 j +1 r +1− j
 υ j ar − j = 0, r = 0,1,2,...
j =0  j 
j =0  j 
r +1
∑
∑
(27)
Finally, one obtains the following formula as a result of equation (27):
υr + 2 =
−1
a0
r
 r  r + 1

r
2
 
υ j +1ar +1− j + Ω
 υ j ar − j  , r = 0,1,2,...
 j = 0  j 

j =0  j 
∑
∑
(28)
As each of coefficients υ r is a combination of coefficients υ0 and υ1, the function U
may be written in the form
V (ξ ) = υ0 P (ξ ) + υ1Q (ξ )
∞
where: P (ξ ) =
∑
r =0
∞
prξ r , Q (ξ ) =
∑q ξ
r
r =0
r
.
(29)
78
S. Kukla, I. Zamojska
Taking into account the conditions (24) in equation (22), one obtains:
u1,0 (ζ ) = −Q (ζ ) w0 (ζ , ζ ) A (ζ ) , u1,1 (ζ ) = P (ζ ) w0 (ζ , ζ ) A (ζ )
(30)
where w0 (ξ , ζ ) = P (ξ )Q′ (ζ ) − Q (ξ ) P′ (ζ ) . Finally, the function G (ξ , ζ ) can be
written in the form
G (ξ , ζ ) = u0,0 (ζ ) P (ξ ) + u0,1 (ζ ) Q (ξ ) + R (ξ , ζ ) H (ξ − ζ ) W (ζ )
(31)
where: W (ζ ) = −1 w0 (ζ , ζ ) A (ζ ) and R (ξ , ζ ) = P (ξ ) Q (ζ ) − P (ζ ) Q (ξ ) .
The coefficients u0,0(ζ) and u0,1(ζ) are determined with the use of boundary conditions. For example, the Green’s functions for clamped-free and free-free rods are
the following:
• a clamped-free rod ( G ξ = 0 = 0, G,ξ
= 0)
ξ =l
G (ξ , ζ ) = W (ζ )  w0 ( l , ζ ) Q (ξ ) Q′ ( l ) + R (ξ , ζ ) H (ξ − ζ ) 
• a free-free rod ( G ,ξ
ξ =0
= 0, G,ξ
ξ =l
(32)
= 0)
G (ξ , ζ ) = W (ζ )  w0 ( l , ζ ) P (ξ ) P′ ( l ) + R (ξ , ζ ) H (ξ − ζ ) 
(33)
5. Numerical example
Let us consider a rod consisting of two segments. First of them is a prismatic
rod with the length l1 and the cross-section area of the second one (with length l2)
is expressed as A2(ξ) = (ξ + 1)4. The rod is clamped at the end ξ = 0 and free at
ξ = L.
The frequency equation (19) for this rod is as follows
[G1 (l1 , l1 ) + µ1G2 (0,0)]1 + η (Ω2 − Λ 2 )G2 (l2 , l2 )  − µ1η (Ω2 − Λ 2 )G2 (l2 , 0)G2 (0, l2 ) = 0
(34)
where the function G2 (ξ , ζ ) is given by formula (33) for i = 2 and G1 (ξ , ζ ) has
the form
G1 (ξ , ζ ) = −Ω−1  cos Ω ( l1 − ζ ) sin Ωξ cos Ω − sin Ω (ξ − ζ ) H (ξ − ζ ) 
(35)
The calculations are performed for various values of the parameter α which
characterizes the non-uniformity of the second segment of the rod.
79
Free Longitudinal Vibrations of Multi-Step Non-Uniform Rods
4.0
6.5
(b)
3.5
6.0
3.0
5.5
2.5
5.0
Ω2
Ω1
(a)
2.0
4.5
α = 1.0
1.5
4.0
α = 0.5
α = -0.5
1.0
3.5
α = -1.0
0.5
3.0
0.0
2.5
0
2
4
6
8
10
12
0
2
4
Λ
8
10
12
8
10
12
Λ
13.0
10.0
(c)
(d)
9.5
12.5
9.0
12.0
8.5
11.5
Ω4
Ω3
6
8.0
11.0
7.5
10.5
7.0
10.0
6.5
9.5
6.0
9.0
0
2
4
6
8
10
12
0
2
4
Λ
6
Λ
Fig. 2. Frequency parameter values Ωi for the first four modes of vibration
as a function of Λ = ω ρ E for a clamped-free rod
with a cross-section area A2 (ξ ) = (αξ + 1)
4
80
S. Kukla, I. Zamojska
A discrete spring-mass element is attached at the free end of the rod. The free
vibration frequency of the isolated spring-mass system is determined by the
parameter Λ = ω ρ E . Natural frequencies of the free longitudinal vibration Ωi
(i = 1,..,4) are numerically calculated as functions of the parameter Λ and they are
presented in Figure 2. The results show that an increase of the frequency of the
isolated spring-mass system causes an increase of free vibration frequencies of the
compound system. It is interesting that in case of second and higher modes there
are such Λ, for which the values of the Ωi do not depend on the non-uniformity
parameter α (the intersection points of the curves on Figures 2(a)-(d)).
6. Conclusions
The solution for free longitudinal vibrations of a rod consisting of n nonuniform segments was obtained with the use of the power series method and the
Green’s function properties. The presented numerical example has shown the influence of the parameter characterizing the attached discrete element on free vibration
frequencies of the system. Although the presented numerical example deals with
non-uniform rods consisting of two segments, the solution can be used for a vibration analysis of rods consisting of an arbitrary number of segments.
References
[1] Bapat C.N., Vibration of rods with uniformly tapered sections, Journal of Sound and Vibration
1995, 185(1), 185-189
[2] Kumar B.M., Sujith R.I., Exact solutions for the longitudinal vibration of non-uniform rods,
Journal of Sound and Vibration 1997, 207(5), 721-729.
[3] Li Q.S., Exact solutions for free longitudinal vibrations of non-uniform rods, Journal of Sound
and Vibration 2000, 234(1), 1-19.
[4] Li Q.S., Free longitudinal vibration analysis of multi-step non-uniform bars based on piecewise
analytical solutions, Engineering Structures 2000, 22, 1205-1215.
[5] Li Q.S., Li G.Q., Liu D.K., Exact solutions for longitudinal vibration of rods coupled by translational springs, International Journal of Mechanical Sciences 2000, 42, 1135-1152.
[6] Li Q.S., Wu J.R., Xu J., Longitudinal vibration of multi-step non-uniform structures with lumped
masses and spring supports, Applied Acoustics 2002, 63, 333-350.
[7] Kukla S., Przybylski J., Tomski L., Longitudinal vibration of rods coupled by translational
springs, Journal of Sound and Vibration 1994, 185, 717-722.