The forces f arising from detached masses can be calculated from the free-body
diagrams. Using Newton’s 2nd law for mass j gives
3
Àf j ¼ m j € x j ; where the negative
sign on f j reflects we are dealing with a reaction forces to those contained in f in
(1.115) and (1.116). Expressing this in matrix form as f ¼ ÀM€ x, where M = diag
(m 1 m 2 ÁÁÁ m n ), one obtains by insertion into (1.115) and (1.116), respectively:
M€ x þ Kx ¼ 0;
ð1:117Þ
AM€ x þ x ¼ 0;
ð1:118Þ
which are two equivalent forms of equations of motion for the system. The stiffness- and flexibility methods, respectively, results in these two forms of equations
of motion. With the stiffness method one has to set up the stiffness matrix K, while
with the flexibility method the flexibility matrix A is required. Typically, for a given
problem, one of these matrices is easier to obtain than the other. Next we outline
and exemplify how each method is used.
1.8.2 The Flexibility Method
The flexibility method will produce at system in the form (1.118) with M = diag(m 1
m 2 ÁÁÁ m n ). The elements a ij of the flexibility matrix A are called flexibility influence
coefficients or just flexibility coefficients. They are determined this way:
a ij is the value of x i when f j =1 and all other f-variables are zero.
To see why this is so it will suffice to write out the components of relation
(1.116) for a system with three degrees of freedom (n = 3):
x 1
x 2
x 3
8
<
:
9
=
;
¼
a 11 a 12 a 13
a 21 a 22 a 23
a 31 a 32 a 33
2
4
3
5
f 1
f 2
f 3
8
<
:
9
=
;
¼
a 11 f 1 þ a 12 f 2 þ a 13 f 3
a 21 f 1 þ a 22 f 2 þ a 23 f 3
a 31 f 1 þ a 32 f 2 þ a 33 f 3
8
<
:
9
=
;
:
ð1:119Þ
As appears one can compute a coefficient such as a 23 as the value of x 2 obtained
when letting f 3 = 1 and f 1 = f 2 = 0, since then the equation for x 2 becomes x 2 = a 23 ;
this is just what the rule dictates.
The computation of flexibility coefficients is often particularly convenient for
beam-type problems where the required displacements due, to a single imposed
force, can be determined from standard lookup tables.
3
Sometimes this is expressed instead as f j ¼ Àm€ x j and then termed d’Alembert’s Principle, but
actually it follows directly from Newton’s 2nd and 3rd law.
38
1 Vibration Basics
diagrams. Using Newton’s 2nd law for mass j gives
3
Àf j ¼ m j € x j ; where the negative
sign on f j reflects we are dealing with a reaction forces to those contained in f in
(1.115) and (1.116). Expressing this in matrix form as f ¼ ÀM€ x, where M = diag
(m 1 m 2 ÁÁÁ m n ), one obtains by insertion into (1.115) and (1.116), respectively:
M€ x þ Kx ¼ 0;
ð1:117Þ
AM€ x þ x ¼ 0;
ð1:118Þ
which are two equivalent forms of equations of motion for the system. The stiffness- and flexibility methods, respectively, results in these two forms of equations
of motion. With the stiffness method one has to set up the stiffness matrix K, while
with the flexibility method the flexibility matrix A is required. Typically, for a given
problem, one of these matrices is easier to obtain than the other. Next we outline
and exemplify how each method is used.
1.8.2 The Flexibility Method
The flexibility method will produce at system in the form (1.118) with M = diag(m 1
m 2 ÁÁÁ m n ). The elements a ij of the flexibility matrix A are called flexibility influence
coefficients or just flexibility coefficients. They are determined this way:
a ij is the value of x i when f j =1 and all other f-variables are zero.
To see why this is so it will suffice to write out the components of relation
(1.116) for a system with three degrees of freedom (n = 3):
x 1
x 2
x 3
8
<
:
9
=
;
¼
a 11 a 12 a 13
a 21 a 22 a 23
a 31 a 32 a 33
2
4
3
5
f 1
f 2
f 3
8
<
:
9
=
;
¼
a 11 f 1 þ a 12 f 2 þ a 13 f 3
a 21 f 1 þ a 22 f 2 þ a 23 f 3
a 31 f 1 þ a 32 f 2 þ a 33 f 3
8
<
:
9
=
;
:
ð1:119Þ
As appears one can compute a coefficient such as a 23 as the value of x 2 obtained
when letting f 3 = 1 and f 1 = f 2 = 0, since then the equation for x 2 becomes x 2 = a 23 ;
this is just what the rule dictates.
The computation of flexibility coefficients is often particularly convenient for
beam-type problems where the required displacements due, to a single imposed
force, can be determined from standard lookup tables.
3
Sometimes this is expressed instead as f j ¼ Àm€ x j and then termed d’Alembert’s Principle, but
actually it follows directly from Newton’s 2nd and 3rd law.
38
1 Vibration Basics