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Sensitivities of mechanical structures to structural parameters
Computers & Structures Vol. 49, No. 3. pp. 557-560. Printed in Great Britain.
1993 0
0045-7949193 $6.00 + 0.00 I993 Pergamon Press Ltd
TECHNICAL NOTE SENSITIVITIES OF MECHANICAL STRUCTURES STRUCTURAL PARAMETERS JIAN-ZHONG
WANG,?
ZHI-CHU
TO
and QING-JIE ZHANG~
HUANGt
TDepartment of Mechanical Engineering and IDepartment of Engineering Mechanics, Wuhan University of Technology, Wuhan 430070, People’s Republic of China (Received 12 June 1992) Abstract-A method is formulated for calculating the sensitivities of natural frequencies and mode shapes directly with respect to structural parameters. The method is valid for discrete systems, and also suitable for distributed structures when combined with the finite element method. Two examples are presented to verify its usefulness.
eqn (3) gives
INTRODUCTION Changing
a mechanical
in order to obtain the most effective change in the dynamic behavior of the structure is obviously an important subject. In this paper, by extending the sensitivity analysis method for natural frequencies and mode shapes with respect to mass and stiffness coefficients of structures, the sensitivities of natural frequencies and mode shapes with respect to structural parameters can be obtained directly. The sensitivities reveal the influences of the structural parameters on the natural frequencies and mode shapes. They provide us with an answer to the question of where to change, for example, where to obtain a maximum shift of a specific natural frequency or mode shape. The result can be used as an indication for the structure dynamics modification.
The vibration characteristics y (e.g. natural frequencies, mode shapes, etc.) of a structure are a function of its physical parameters x, (e.g. mass, stiffness, damping, structural data, etc.) of the structure. The relative sensitivities of y with respect to x, (i = I, 2,. , m) are defined as AYIY
SLy 1x,] = lim ~~_---, AX,-oAx,/x,
x, ay Y ax,
+ Kl{xI = VI.
[9m41[91=
PI
(6)
[9IT[fa91=
VI.
(7)
where [9] is a modal matrix, [I] is an identity matrix, [A] is a diagonal matrix composed of eigenvalues of the system, and 1, and {cp,}are the r th eigenvalue and its corresponding normalized eigenvector, respectively. The superscript T denotes the transpose of a matrix. Let P, be a structural parameter of the system, the partial differentiation of eqn (4) with respect to P, indicates + Fl gl=
1
t34{9,i , gL
+~,~~9J+wl
(8)
Premultiplying eqn (8) by (9,)’ and making use of eqns (5) and (6) results in
(
(1)
(9)
1
Due to I, = wf, eqn (9) can be written as
am,
1
ap=2w{9,}T
,
r
50$39.~~
(
(10)
I
where o, is the rth natural circular frequency of the system. Then, the relative sensitivity of natural frequency with respect to P, is directly obtained from eqns (1) and (10) as
(2)
where [M] is mass matrix and [K] is stiffness matrix. It creates an eigenproblem in the form WI(x) = mfl{xl.
(3
~,{cp,~Wl
$={p,}’ ag-n,3(9,).
The sensitivities expressed in eqn (1) are normalized sensitivities of the first-order. The equation of motion for an undamped linear dynamical system can be written as Pfl{fI
{cp,)Wl=
WI Y&9,1
FIRST-ORDER SENSITIVITIES OF NATURAL FREQUENCIES AND MODE SHAPES
(4)
Kl{9,) = m41{9,~
structure
(3)
In the case when [M] and [K] are symmetric matrices and the system has distinct eigenvalues and eigenvectors, then
s[w,,P,1=~{9,}r
r
gL:ag
(
I
$1
(cp,}.
(11)
Premultiplying eqn (8) by {cp,}‘, where s # r, and making use of eqns (5) and (6), yields
558
Technical
Note P, is obtained
directly
from eqns (I), (13) and (17) as
S[9~,lP,l=~,~,n,,{~,i. Fig. 1. Four degrees-of-freedom
spring-mass
k=1,2.....n.
(18)
system where
the vector combination
d{cp,}/aP, may be expressed of eigenvectors. Thus, let
and by substituting
a IS=&Jr
r
as
a
linear
eqn (13) into (12) one has
!$1,%
I
(
{cp,},
#
s#r.
The sensitivities expressed in eqns (11) and (18) are normalized sensitivities of the first-order. An important computational task in eqns (11) and (19) is the calculation of a[K]/aP, and c?[M]/~P,. If the structural model contains only elements whose stiffness and mass matrices are proportional to P,, a[K]/aP, and a[M]/cYP, are constant matrices. But for elements that have bending stiffnesses such as beams and plates, the stiffness and mass matrices are nonlinear functions of P,, and the d [K]/c?P, and a[M]/cYP, are not easily evaluated. The preferred approach is to replace cY[K]/aP, and c?[M]/aP, by A[K]/AP, and A[M]/AP,, respectively. So long as BP, is small enough, this approximate calculation is accurate enough. A[K]/AP, and A[M]/AP, can be obtained by the finite element method.
(14)
z>
In the case of s = r, a_ = G(,,can be found by differentiating {cp,]Wl{cp,) with respect
1
to P, which results in
2{rp,)r[MlF+ Substituting
=
i9,)‘F
{9,} =o.
(16)
eqn (13) into (16) gives EXAMPLE I
Then, the relative sensitivity
A discrete system composed of four lumped masses and four massless springs is shown in Fig. 1. Each mass is a ball, and the springs are cylindrical. The structural parameters of
of mode shape with respect to
Table
1. Structural
parameters
of the system
D,=D,=D,=D,=O.O2m m, = p,nD;
k, = G,dj/(8D;n,)
m,=pzzD:
k, = G,d;/(8D;n,)
d, = d, = d, = d., = 0.002 m G, = G, = G, = G, = 1.92 x IO5MPa
m, = P~zD:
k, = G,d:l(8D:n,)
n, = 60, nz = 30, n3 = 20. n4 = I5
nz4= panDj
k, = G,d:/(8D34n,)
p, = 3.9789 x IO“kg/m’ P2 = P3 = 2P,. P4 = 3Pl
D is the diameter of the ball and spring, d is the diameter of the steel wire of spring, G is shear modulus of the material, n is the number of the spring turns, and p is material density of the ball.
Table 2. System characteristics
lo Stiffness
matrix
Natural o, = w2 = oj = w., =
frequencies 13.294 29.660 41.079 55.882
0
k,
-k,
0
0
k, +k,
-k,
0
-k, 0
k,+k,
0
d
!
(N/m)
-k, 0 I
0
-k,
-k,
-800 8000
-1600 -800 24000
--2400 4000 1600 0
-2400 5600 0
=
k, + k, I
(rad/sec)
Mode shapes {9,}r={ 0.5901 {cpz}’ = { -0.6780 {9,}‘= { 0.4314 {(Pi}‘= { -0.0876
0.4601 0.0683 -0.4786 0.2354
(kg-“‘)
0.2927 0.3658 0.0763 -0.5241
0.1387) 0.2968) 0.3390) 0.3343}
Technical Note
559
Table 3. S[w, 1P,]: sensitivities of natural frequencies with respect to parameters of the system
-0.6371 -1.1056 -0.7400 -0.6091 0.1530 0.5074 0.6441 0.6967 -0.0383 -0.1268 -0.1610 -0.1742 -0.1741 -0.2117 -0.0857 -0.0289
D, 4 D3 D4 4 d2 d3 4 4 n2 n3 4 PI P2
Pt
P4
- 1.4365 -0.2555 - 0.4209 -0.8771 1.0130 0.3219 0.0260 0.6409 -0.2532 - 0.0805 -0.0065 -0.1602 - 0.2298 - 0.0047 -0.1338 -0.1321
-0.8422 - 1.1251 -0.1647 -0.8440 0.7852 0.5839 0.1963 0.4358 -0.1028 -0.1460 -0.0491 -0.1090 -0.0931 -0.2291 -0.0058 -0.1724
-0.0516 -0.6096 - 1.6735 -0.6747 0.0535 0.5911 1.1326 0.2290 -0.0134 -0.1478 -0.2832 -0.0573 -0.0038 -0.0554 -0.2747 -0.1676
Table 4. System parameters and natural frequencies No. of cross-section Parameters (mm)
1
2
3
4
5
6
7
8
9
10
II
12
13
6
1.0 30.0 21.0
1.0 35.0 24.0
1.0 40.0 27.0
1.0 45.0 30.0
1.0 50.0 33.0
1.0 42.0 21.0
1.0 36.0 21.0
1.0 30.0 21.0
1.0 24.0 21.0
1.0 24.0 21.0
1.0 30.0 21.0
1.0
1.0 42.0 21.0
h b
M (mass) = 10.0 kg m, (mass) = 3.5 kg E (elastic modulus) = 2. I x IO4MPa G (shear modulus) = 8.1 x 104MPa p (material density) = 7.8 x 10m3kg/cm).
o, = w2 = o3 = o4 =
138.73 rad/sec 175.36 rad/sec 219.66 rad/sec 254.41 rad/sec
the balls and springs are represented in Table 1. The mass and stiffness matrices accompany with natural frequencies and mode shapes are given in Table 2. By making use of eqn (1 l), the sensitivities of natural frequencies with respect to parameters of the system are calculated, and the results are given in Table 3. There are many parameters that can be chosen for modification and the one which gives the most pronounced effect with the least effort will be preferred in general. The choice may be judged by examining the values of sensitivities and a larger sensitivity will result in a smaller modification. From the sensitivities given in Table 3, it may be concluded that the natural frequencies will shift downwards by
Table 5. S[o, 1Pi]: sensitivities of the first natural frequency with respect to structural parameters No. of cross-section 1 2 3 4 5 6 7 8 9 10 11 12 13
W,
I p, I
-0.09 -0.18 -0.25 -0.34 -0.42 -0.36 -0.28 -0.22 -0.19 -0.18 -0.23 -0.29 -0.35
a4
Ip21
+0.46 +0.49 +0.52 +0.56 +0.58 +0.87 +0.82 +0.78 +0.71 +0.71 +0.79 +0.82 +0.88
a4
Ip21
+I.08 +1.17 +1.26 f1.42 +1.54 +0.44 +0.41 +0.38 f0.35 +0.36 +0.38 +0.42 +0.44
36.0 21.0
reducing the d in points 14, or by adding the D, n and p in points 14. If the d or D or n or p is changed, it is obvious that the d4 or D, or n4 or p2, respectively, is the best as a choice for obtaining a natural frequency shift of the first mode. The sensitivities of the other mode natural frequencies indicate the sensitive points for modification similarly. EXAMPLE 2
The tripod shown in Fig. 2(a) is a machine gun mount. The cross-section of each leg is shown in Fig. 2(b). Referring to Fig. 2, the system parameters and the first four natural frequencies obtained by finite element method are given in Table 4. Let P, =6, P2= h + b and P3 = h/b be the structural parameters which will be changed. By making use of eqn (11) and the finite element method, the sensitivities of the first natural frequency o, with respect to P, (i = 1,2,3) are obtained. The results are given in Table 5.
Fig. 2. Structure of a machine gun tripod.
560
Technical
From the sensitivities given in Table 5, it may be concluded that the first natural frequency will shift downward by adding the P, =6, or by reducing the P2 = (h f h) or P, = h/b. The P, in points I-5 is clearly the sensitive parameter for modification of the tripod.
Note REFERENCES 1. H. M. A. Delman
and R. Haftka, Sensitivity analysis of structural systems. AIAA Jnl 24, 823-832
2.
3. CONCLUSION
The method presented in this paper is valid for calculating the relative sensitivities of mechanical structures directly to structural parameters. Two examples verify its usefulness not only for discrete systems but also for distributed systems (real structures). A further study to be carried out is to reduce the amount of work for calculating the A[K]/AP, and
A[WIW.
4.
5.
discrete (1986). P. Vanhonacker, Differential and difference sensitivities of natural frequencies and mode shapes of mechanical structures. AIAA Jnl 18, 1511-1514 (1980). C. S. Rudisill, Derivatives of eigenvalues and eigenvectors for a general matrix. AIAA Jnl 12, 721-722 (1974). C. Rudisill, Numerical method for evaluating the derivatives of eigenvalues and eigenvectors. AIAA Jnl 13, 834-836 (1975). R. T. Haftka and D. S. Malkus, Calculation of sensitivity derivatives in thermal problems by finite differences. Int. J. Namer. Merh. Engng 17, 181 I-1821 (1981).
1993 0
0045-7949193 $6.00 + 0.00 I993 Pergamon Press Ltd
TECHNICAL NOTE SENSITIVITIES OF MECHANICAL STRUCTURES STRUCTURAL PARAMETERS JIAN-ZHONG
WANG,?
ZHI-CHU
TO
and QING-JIE ZHANG~
HUANGt
TDepartment of Mechanical Engineering and IDepartment of Engineering Mechanics, Wuhan University of Technology, Wuhan 430070, People’s Republic of China (Received 12 June 1992) Abstract-A method is formulated for calculating the sensitivities of natural frequencies and mode shapes directly with respect to structural parameters. The method is valid for discrete systems, and also suitable for distributed structures when combined with the finite element method. Two examples are presented to verify its usefulness.
eqn (3) gives
INTRODUCTION Changing
a mechanical
in order to obtain the most effective change in the dynamic behavior of the structure is obviously an important subject. In this paper, by extending the sensitivity analysis method for natural frequencies and mode shapes with respect to mass and stiffness coefficients of structures, the sensitivities of natural frequencies and mode shapes with respect to structural parameters can be obtained directly. The sensitivities reveal the influences of the structural parameters on the natural frequencies and mode shapes. They provide us with an answer to the question of where to change, for example, where to obtain a maximum shift of a specific natural frequency or mode shape. The result can be used as an indication for the structure dynamics modification.
The vibration characteristics y (e.g. natural frequencies, mode shapes, etc.) of a structure are a function of its physical parameters x, (e.g. mass, stiffness, damping, structural data, etc.) of the structure. The relative sensitivities of y with respect to x, (i = I, 2,. , m) are defined as AYIY
SLy 1x,] = lim ~~_---, AX,-oAx,/x,
x, ay Y ax,
+ Kl{xI = VI.
[9m41[91=
PI
(6)
[9IT[fa91=
VI.
(7)
where [9] is a modal matrix, [I] is an identity matrix, [A] is a diagonal matrix composed of eigenvalues of the system, and 1, and {cp,}are the r th eigenvalue and its corresponding normalized eigenvector, respectively. The superscript T denotes the transpose of a matrix. Let P, be a structural parameter of the system, the partial differentiation of eqn (4) with respect to P, indicates + Fl gl=
1
t34{9,i , gL
+~,~~9J+wl
(8)
Premultiplying eqn (8) by (9,)’ and making use of eqns (5) and (6) results in
(
(1)
(9)
1
Due to I, = wf, eqn (9) can be written as
am,
1
ap=2w{9,}T
,
r
50$39.~~
(
(10)
I
where o, is the rth natural circular frequency of the system. Then, the relative sensitivity of natural frequency with respect to P, is directly obtained from eqns (1) and (10) as
(2)
where [M] is mass matrix and [K] is stiffness matrix. It creates an eigenproblem in the form WI(x) = mfl{xl.
(3
~,{cp,~Wl
$={p,}’ ag-n,3(9,).
The sensitivities expressed in eqn (1) are normalized sensitivities of the first-order. The equation of motion for an undamped linear dynamical system can be written as Pfl{fI
{cp,)Wl=
WI Y&9,1
FIRST-ORDER SENSITIVITIES OF NATURAL FREQUENCIES AND MODE SHAPES
(4)
Kl{9,) = m41{9,~
structure
(3)
In the case when [M] and [K] are symmetric matrices and the system has distinct eigenvalues and eigenvectors, then
s[w,,P,1=~{9,}r
r
gL:ag
(
I
$1
(cp,}.
(11)
Premultiplying eqn (8) by {cp,}‘, where s # r, and making use of eqns (5) and (6), yields
558
Technical
Note P, is obtained
directly
from eqns (I), (13) and (17) as
S[9~,lP,l=~,~,n,,{~,i. Fig. 1. Four degrees-of-freedom
spring-mass
k=1,2.....n.
(18)
system where
the vector combination
d{cp,}/aP, may be expressed of eigenvectors. Thus, let
and by substituting
a IS=&Jr
r
as
a
linear
eqn (13) into (12) one has
!$1,%
I
(
{cp,},
#
s#r.
The sensitivities expressed in eqns (11) and (18) are normalized sensitivities of the first-order. An important computational task in eqns (11) and (19) is the calculation of a[K]/aP, and c?[M]/~P,. If the structural model contains only elements whose stiffness and mass matrices are proportional to P,, a[K]/aP, and a[M]/cYP, are constant matrices. But for elements that have bending stiffnesses such as beams and plates, the stiffness and mass matrices are nonlinear functions of P,, and the d [K]/c?P, and a[M]/cYP, are not easily evaluated. The preferred approach is to replace cY[K]/aP, and c?[M]/aP, by A[K]/AP, and A[M]/AP,, respectively. So long as BP, is small enough, this approximate calculation is accurate enough. A[K]/AP, and A[M]/AP, can be obtained by the finite element method.
(14)
z>
In the case of s = r, a_ = G(,,can be found by differentiating {cp,]Wl{cp,) with respect
1
to P, which results in
2{rp,)r[MlF+ Substituting
=
i9,)‘F
{9,} =o.
(16)
eqn (13) into (16) gives EXAMPLE I
Then, the relative sensitivity
A discrete system composed of four lumped masses and four massless springs is shown in Fig. 1. Each mass is a ball, and the springs are cylindrical. The structural parameters of
of mode shape with respect to
Table
1. Structural
parameters
of the system
D,=D,=D,=D,=O.O2m m, = p,nD;
k, = G,dj/(8D;n,)
m,=pzzD:
k, = G,d;/(8D;n,)
d, = d, = d, = d., = 0.002 m G, = G, = G, = G, = 1.92 x IO5MPa
m, = P~zD:
k, = G,d:l(8D:n,)
n, = 60, nz = 30, n3 = 20. n4 = I5
nz4= panDj
k, = G,d:/(8D34n,)
p, = 3.9789 x IO“kg/m’ P2 = P3 = 2P,. P4 = 3Pl
D is the diameter of the ball and spring, d is the diameter of the steel wire of spring, G is shear modulus of the material, n is the number of the spring turns, and p is material density of the ball.
Table 2. System characteristics
lo Stiffness
matrix
Natural o, = w2 = oj = w., =
frequencies 13.294 29.660 41.079 55.882
0
k,
-k,
0
0
k, +k,
-k,
0
-k, 0
k,+k,
0
d
!
(N/m)
-k, 0 I
0
-k,
-k,
-800 8000
-1600 -800 24000
--2400 4000 1600 0
-2400 5600 0
=
k, + k, I
(rad/sec)
Mode shapes {9,}r={ 0.5901 {cpz}’ = { -0.6780 {9,}‘= { 0.4314 {(Pi}‘= { -0.0876
0.4601 0.0683 -0.4786 0.2354
(kg-“‘)
0.2927 0.3658 0.0763 -0.5241
0.1387) 0.2968) 0.3390) 0.3343}
Technical Note
559
Table 3. S[w, 1P,]: sensitivities of natural frequencies with respect to parameters of the system
-0.6371 -1.1056 -0.7400 -0.6091 0.1530 0.5074 0.6441 0.6967 -0.0383 -0.1268 -0.1610 -0.1742 -0.1741 -0.2117 -0.0857 -0.0289
D, 4 D3 D4 4 d2 d3 4 4 n2 n3 4 PI P2
Pt
P4
- 1.4365 -0.2555 - 0.4209 -0.8771 1.0130 0.3219 0.0260 0.6409 -0.2532 - 0.0805 -0.0065 -0.1602 - 0.2298 - 0.0047 -0.1338 -0.1321
-0.8422 - 1.1251 -0.1647 -0.8440 0.7852 0.5839 0.1963 0.4358 -0.1028 -0.1460 -0.0491 -0.1090 -0.0931 -0.2291 -0.0058 -0.1724
-0.0516 -0.6096 - 1.6735 -0.6747 0.0535 0.5911 1.1326 0.2290 -0.0134 -0.1478 -0.2832 -0.0573 -0.0038 -0.0554 -0.2747 -0.1676
Table 4. System parameters and natural frequencies No. of cross-section Parameters (mm)
1
2
3
4
5
6
7
8
9
10
II
12
13
6
1.0 30.0 21.0
1.0 35.0 24.0
1.0 40.0 27.0
1.0 45.0 30.0
1.0 50.0 33.0
1.0 42.0 21.0
1.0 36.0 21.0
1.0 30.0 21.0
1.0 24.0 21.0
1.0 24.0 21.0
1.0 30.0 21.0
1.0
1.0 42.0 21.0
h b
M (mass) = 10.0 kg m, (mass) = 3.5 kg E (elastic modulus) = 2. I x IO4MPa G (shear modulus) = 8.1 x 104MPa p (material density) = 7.8 x 10m3kg/cm).
o, = w2 = o3 = o4 =
138.73 rad/sec 175.36 rad/sec 219.66 rad/sec 254.41 rad/sec
the balls and springs are represented in Table 1. The mass and stiffness matrices accompany with natural frequencies and mode shapes are given in Table 2. By making use of eqn (1 l), the sensitivities of natural frequencies with respect to parameters of the system are calculated, and the results are given in Table 3. There are many parameters that can be chosen for modification and the one which gives the most pronounced effect with the least effort will be preferred in general. The choice may be judged by examining the values of sensitivities and a larger sensitivity will result in a smaller modification. From the sensitivities given in Table 3, it may be concluded that the natural frequencies will shift downwards by
Table 5. S[o, 1Pi]: sensitivities of the first natural frequency with respect to structural parameters No. of cross-section 1 2 3 4 5 6 7 8 9 10 11 12 13
W,
I p, I
-0.09 -0.18 -0.25 -0.34 -0.42 -0.36 -0.28 -0.22 -0.19 -0.18 -0.23 -0.29 -0.35
a4
Ip21
+0.46 +0.49 +0.52 +0.56 +0.58 +0.87 +0.82 +0.78 +0.71 +0.71 +0.79 +0.82 +0.88
a4
Ip21
+I.08 +1.17 +1.26 f1.42 +1.54 +0.44 +0.41 +0.38 f0.35 +0.36 +0.38 +0.42 +0.44
36.0 21.0
reducing the d in points 14, or by adding the D, n and p in points 14. If the d or D or n or p is changed, it is obvious that the d4 or D, or n4 or p2, respectively, is the best as a choice for obtaining a natural frequency shift of the first mode. The sensitivities of the other mode natural frequencies indicate the sensitive points for modification similarly. EXAMPLE 2
The tripod shown in Fig. 2(a) is a machine gun mount. The cross-section of each leg is shown in Fig. 2(b). Referring to Fig. 2, the system parameters and the first four natural frequencies obtained by finite element method are given in Table 4. Let P, =6, P2= h + b and P3 = h/b be the structural parameters which will be changed. By making use of eqn (11) and the finite element method, the sensitivities of the first natural frequency o, with respect to P, (i = 1,2,3) are obtained. The results are given in Table 5.
Fig. 2. Structure of a machine gun tripod.
560
Technical
From the sensitivities given in Table 5, it may be concluded that the first natural frequency will shift downward by adding the P, =6, or by reducing the P2 = (h f h) or P, = h/b. The P, in points I-5 is clearly the sensitive parameter for modification of the tripod.
Note REFERENCES 1. H. M. A. Delman
and R. Haftka, Sensitivity analysis of structural systems. AIAA Jnl 24, 823-832
2.
3. CONCLUSION
The method presented in this paper is valid for calculating the relative sensitivities of mechanical structures directly to structural parameters. Two examples verify its usefulness not only for discrete systems but also for distributed systems (real structures). A further study to be carried out is to reduce the amount of work for calculating the A[K]/AP, and
A[WIW.
4.
5.
discrete (1986). P. Vanhonacker, Differential and difference sensitivities of natural frequencies and mode shapes of mechanical structures. AIAA Jnl 18, 1511-1514 (1980). C. S. Rudisill, Derivatives of eigenvalues and eigenvectors for a general matrix. AIAA Jnl 12, 721-722 (1974). C. Rudisill, Numerical method for evaluating the derivatives of eigenvalues and eigenvectors. AIAA Jnl 13, 834-836 (1975). R. T. Haftka and D. S. Malkus, Calculation of sensitivity derivatives in thermal problems by finite differences. Int. J. Namer. Merh. Engng 17, 181 I-1821 (1981).