Vibrational modes of multispan EulerBernoulli beams
through KrylovDunction functions
Rouben Rostamian
20200719
Note: Maple defines the imaginary unit . We want to use the
symbol as the beam's crosssectional moment of inertia.
Therefore we redefine the imaginary unit (for which we have no
use) as and free up the symbol for our use.
> 
interface(imaginaryunit=II):


The EulerBernoulli beam equation
.


We wish to determine the natural modes of vibration of
a possibly multispan EulerBernoulli beam.
Separate the variables by setting . We get

whence
.
Let . Then
and
The idea behind the KrylovDuncan technique is to express
in terms an alternative (and equivalent) set of basis
functions through ,, as
,
where the functions through are defined in the next section.
In some literature the symbols , are used for these
functions but I find it more sensible to use the indexed function
notation.
The KrylovDuncan approach is particularly effective in formulating
and finding a multispan beam's natural modes of vibration.


The KrylovDuncan functions


The K[i](x) defined by this proc evaluates to the th
KrylovDuncan function.
Normally the index will be in the set, however the proc is
set up to accept any integer index (positive or negative). The proc evaluates
the index modulo 4 to bring the index into the set . For
instance, K[5](x) and K[3](x)i are equivalent to K[1](x) .
> 
K := proc(x)
local n := op(procname);
if not type(n, integer) then
return 'procname'(args);
else
n := 1 + modp(n1,4); # reduce n modulo 4
end if;
if n=1 then
(cosh(x) + cos(x))/2;
elif n=2 then
(sinh(x) + sin(x))/2;
elif n=3 then
(cosh(x)  cos(x))/2;
elif n=4 then
(sinh(x)  sin(x))/2;
else
error "shouldn't be here!";
end if;
end proc:

Here are the KrylovDuncan basis functions:
> 
seq(print(cat(`K__`,i)(x) = K[i](x)), i=1..4);

and here is what they look like. All grow exponentially for large
but are significantly different near the origin.
> 
plot([K[i](x) $i=1..4], x=Pi..Pi,
color=["red","Green","blue","cyan"],
thickness=2,
legend=['K[1](x)', 'K[2](x)', 'K[3](x)', 'K[4](x)']);

The cyclic property of the derivatives:
We have . Let's verify that:
> 
diff(K[i](x),x)  K[i1](x) $i=1..4;

The fourth derivative of each function equals itself. This is a consequence of the cyclic property:
> 
diff(K[i](x), x$4)  K[i](x) $ i=1..4;

The essential property of the KrylovDuncan basis function is that their
zeroth through third derivatives at form a basis for :
> 
seq((D@@n)(K[1])(0), n=0..3);
seq((D@@n)(K[2])(0), n=0..3);
seq((D@@n)(K[3])(0), n=0..3);
seq((D@@n)(K[4])(0), n=0..3);

As noted earlier, in the case of a singlespan beam, the modal shapes
are expressed as
.
Then, due to the cyclic property of the derivatives of the KrylovDuncan
functions, we see that:
.
.
.
Let us note, in particular, that
,
,
,
.


A general approach for solving multispan beams


In a multispan beam, we write for the deflection of the th span, where
and where is the span's length. The coordinate indicates the
location within the span, with corresponding to the span's left endpoint.
Thus, each span has its own coordinate system.
We assume that the interface of the two adjoining spans is supported on springs
which (a) resist transverse displacement proportional to the displacement (constant of
proportionality of (d for displacement), and (b) resist rotation proportional to the
slope (constant of proportionality of (t for torsion or twist). The spans are numbered
from left to right. The interface conditions between spans and +1 are
1. 
The displacements at the interface match:
.

2. 
The slopes at the interface match
(0).

3. 
The difference of the moments just to the left and just to the right of the
support is due to the torque exerted by the torsional spring:

4. 
The difference of the shear forces just to the left and just to the right of the
support is due to the force exerted by the linear spring:

The special case of a pinned support corresponds to and .
In that case, condition 3 above implies that ,
and condition 4 implies that
Let us write the displacements and in terms of the KrylovDuncan
functions as:
Then applying the cyclic properties of the KrylovDuncan functions described
earlier, the four interface conditions translate to the following system of four
equations involving the eight coefficients .
,
which we write as a matrix equation
.
That coefficient matrix plays a central role in solving
for modal shapes of multispan beams. Let's call it .
Note that the value of enters that matrix only in combinations with
and . Therefore we introduce the new symbols
, .
The following proc generates the matrix . The parameters and
are optional and are assigned the default values of infinity and zero, which
corresponds to a pinned support.
The % sign in front of each Krylov function makes the function inert, that is, it
prevents it from expanding into trig functions. This is so that we can
see, visually, what our expressions look like in terms of the functions. To
force the evaluation of those inert function, we will apply Maple's value function,
as seen in the subsequent demos.
> 
M_interface := proc(mu, L, {Kd:=infinity, Kt:=0})
local row1, row2, row3, row4;
row1 := %K[1](mu*L), %K[2](mu*L), %K[3](mu*L), %K[4](mu*L), 1, 0, 0, 0;
row2 := %K[4](mu*L), %K[1](mu*L), %K[2](mu*L), %K[3](mu*L), 0, 1, 0, 0;
row3 := %K[3](mu*L), %K[4](mu*L), %K[1](mu*L), %K[2](mu*L), 0, Kt/mu, 1, 0;
if Kd = infinity then
row4 := 0, 0, 0, 0, 1, 0, 0, 0 ;
else
row4 := %K[2](mu*L), %K[3](mu*L), %K[4](mu*L), %K[1](mu*L), Kd/mu^3, 0, 0, 1;
end if:
return < <row1>  <row2>  <row3>  <row4> >^+;
end proc:

Here is the interface matrix for a pinned support:
And here is the interface matrix for a general springy support:
> 
M_interface(mu, L, 'Kd'=a, 'Kt'=b);

Note: In Maple's Java interface, inert quantities are shown in gray.
Note: The in this matrix is the length of the span to the left of the interface.
Recall that it is , not , in the derivation that leads to that matrix.
In a beam consisting of spans, we write the th span's deflection as
Solving the beam amounts to determining the unknowns
which we order as
At each of the interface supports we have a set of four equations as derived
above, for a total of equations. Additionally, we have four usersupplied
boundary conditions  two at the extreme left and two at the extreme right of the
overall beam. Thus, altogether we have equations which then we solve for the
unknown coefficients .
The usersupplied boundary conditions at the left end are two equations, each in the
form of a linear combination of the coefficients . We write for the
coefficient matrix of that set of equations. Similarly, the usersupplied boundary
conditions at the right end are two equations, each in the form of a linear combination
of the coefficients . We write for the coefficient matrix of
that set of equations. Putting these equations together with those obtained at the interfaces,
we get a linear set of equations represented by a matrix which can be assembled
easily from the matrices , and . In the case of a 4span beam the
assembled matrix looks like this:
The pattern generalizes to any number of spans in the obvious way.
For future use, here we record a few frequently occurring and matrices.
> 
M_left_pinned := <
1, 0, 0, 0;
0, 0, 1, 0 >;

> 
M_right_pinned := (mu,L) > <
%K[1](mu*L), %K[2](mu*L), %K[3](mu*L), %K[4](mu*L);
%K[3](mu*L), %K[4](mu*L), %K[1](mu*L), %K[2](mu*L) >;

> 
M_left_clamped := <
1, 0, 0, 0;
0, 1, 0, 0 >;

> 
M_right_clamped := (mu,L) > <
%K[1](mu*L), %K[2](mu*L), %K[3](mu*L), %K[4](mu*L);
%K[4](mu*L), %K[1](mu*L), %K[2](mu*L), %K[3](mu*L) >;

> 
M_left_free := <
0, 0, 1, 0;
0, 0, 0, 1 >;

> 
M_right_free := (mu,L) > <
%K[3](mu*L), %K[4](mu*L), %K[1](mu*L), %K[2](mu*L);
%K[2](mu*L), %K[3](mu*L), %K[4](mu*L), %K[1](mu*L) >;

The following proc builds the overall matrix in the general case. It takes
two or three arguments. The first two arguments are the matrices
which are called and in the discussion above. If the beam
consists of a single span, that's all the information that need be supplied.
There is no need for the third argument.
In the case of a multispan beam, in the third argument we supply the
list of the interface matrices , as in , listed in order
of the supports, from left to right. An empty list is also
acceptable and is interpreted as having no internal supports,
i.e., a singlespan beam.
> 
build_matrix := proc(left_bc::Matrix(2,4), right_bc::Matrix(2,4), interface_matrices::list)
local N, n, i, M;
# n is the number of internal supports
n := 0;
# adjust n if a third argument is supplied
if _npassed = 3 then
n := nops(interface_matrices);
if n > 0 then
for i from 1 to n do
if not type(interface_matrices[i], 'Matrix(4,8)') then
error "expected a 4x8 matrix for element %1 in the list of interface matrices", i;
end if;
end do;
end if;
end if;
N := n + 1; # number of spans
M := Matrix(4*N);
M[1..2, 1..4] := left_bc;
for i from 1 to n do
M[4*i1..4*i+2, 4*i3..4*i+4] := interface_matrices[i];
end do;
M[4*N1..4*N, 4*N3..4*N] := right_bc;
return M;
end proc:

For instance, for a singlespan cantilever beam of length we get the following matrix:
> 
build_matrix(M_left_clamped, M_right_free(mu,L));

For a twospan beam with with span lengths of and , and all three
supports pinned, we get the following matrix:
> 
build_matrix(M_left_pinned, M_right_pinned(mu,L[2]), [M_interface(mu, L[1])]);

The matrix represents a homogeneous linear system (i.e., the righthand side vector
is zero.) To obtain a nonzero solution, we set the determinant of equal to zero.
That gives us a generally transcendental equation in the single unknown . Normally
the equation has infinitely many solutions. We call these
Remark: In the special case of pinned supports at the interfaces, that is, when
, the matrix depends only on the span lengths .
It is independent of the parameters that enter the EulerBernoulli
equations. The frequencies , however, depend on those parameters.
This proc plots the calculated modal shape corresponding to the eigenvalue .
The params argument is a set of equations which define the numerical values
of all the parameters that enter the problem's description, such as the span
lengths.
It is assumed that in a multispan beam, the span lengths are named etc.,
and in a singlespan beam, the length is named .
> 
plot_beam := proc(M::Matrix,mu::realcons, params::set)
local null_space, N, a_vals, i, j, A, B, P;
eval(M, params);
eval(%, :mu=mu);
value(%); #print(%);
null_space := NullSpace(%); #print(%);
if nops(null_space) <> 1 then
error "Calculation failed. Increasing Digits and try again";
end if;
N := Dimension(M)[1]/4; # number of spans
a_vals := convert([seq(seq(a[i,j], j=1..4), i=1..N)] =~ null_space[1], list);
if N = 1 then
eval(add(a[1,j]*K[j](mu*x), j=1..4), a_vals);
P[1] := plot(%, x=0..eval(L,params));
else
A := 0;
B := 0;
for i from 1 to N do
B := A + eval(L[i], params);
eval(add(a[i,j]*K[j](mu*x), j=1..4), a_vals);
eval(%, x=xA):
P[i] := plot(%, x=A..B);
A := B;
end do;
unassign('i');
end if;
plots:display([P[i] $i=1..N]);
end proc:



A singlespan pinnedpinned beam


Here we calculate the natural modes of vibration of a single span
beam, pinned at both ends. The modes are of the form
The matrix is:
> 
M := build_matrix(M_left_pinned, M_right_pinned(mu,L));

The characteristic equation:
> 
Determinant(M);
eq := simplify(value(%)) = 0;

> 
solve(eq, mu, allsolutions);

We conclude that the eigenvalues are .
A nontrivial solution of the system is in the nullspace of :
> 
eval(value(M), mu=n*Pi/L) assuming n::integer;
N := NullSpace(%);

Here are the weights that go with the Krylov functions:
> 
a_vals := convert([a[1,j] $j=1..4] =~ N[1], set);

and here is the deflection:
> 
add(a[1,j]*K[j](mu*x), j=1..4);
eval(%, a_vals); # plug in the a_vals calculated above
eval(%, mu=n*Pi/L); # assert that n is an integer

We see that the shape functions are simple sinusoids.


A singlespan freefree beam


Here we calculate the natural modes of vibration of a single span
beam, free at both ends. The modes are of the form
.
The reasoning behind the calculations is very similar to that in the
previous section, therefore we don't comment on many details.
> 
M := build_matrix(M_left_free, M_right_free(mu,L));

The characteristic equation:
> 
Determinant(M);
simplify(value(%)) = 0;
eq_tmp := isolate(%, cos(L*mu));

Let . Then the characteristic equation takes the form
> 
eq := algsubs(L*mu=lambda, eq_tmp);

Here are the graphs of the two sides of the characteristic equation:
> 
plot([lhs,rhs](eq), lambda=0..4*Pi, color=["red","Green"]);

The first three roots are:
> 
lambda__1, lambda__2, lambda__3 :=
fsolve(eq, lambda=Pi/2..4*Pi, maxsols=3);

> 
mu__1, mu__2, mu__3 := (lambda__1, lambda__2, lambda__3) /~ eval(L, params);

> 
plots:display([
plot_beam(M, mu__1, params),
plot_beam(M, mu__2, params),
plot_beam(M, mu__3, params)],
color=["red","Green","blue"],
legend=[mode1, mode2, mode3]);



A singlespan clampedfree cantilever


We have a cantilever beam of length . It is clamped at the
left end, and free at the right end.
> 
M := build_matrix(M_left_clamped, M_right_free(mu,L));

> 
Determinant(M);
simplify(value(%)) = 0;
eq_tmp := isolate(%, cos(L*mu));

Let . Then the characteristic equation takes the form
> 
eq := algsubs(L*mu=lambda, eq_tmp);

Here are the graphs of the two sides of the characteristic equation:
> 
plot([lhs,rhs](eq), lambda=0..3*Pi, color=["red","Green"]);

> 
lambda__1, lambda__2, lambda__3 :=
fsolve(eq, lambda=Pi/2..3*Pi, maxsols=3);

> 
mu__1, mu__2, mu__3 := (lambda__1, lambda__2, lambda__3) /~ eval(L, params);

> 
plots:display([
plot_beam(M, mu__1, params),
plot_beam(M, mu__2, params),
plot_beam(M, mu__3, params)],
color=["red","Green","blue"],
legend=[mode1, mode2, mode3]);



A dualspan pinnedpinnedfree cantilever beam


We have a twospan beam of span lengths and , with the left end of the
first span pinned, the right end of the second span free, and the interface
between the spans on a pinned support. .
> 
M := build_matrix(
M_left_pinned,
M_right_free(mu,L[2]),
[ M_interface(mu,L[1])] );

The characteristic equation:
> 
Determinant(M);
eq_tmp1 := simplify(value(%)) = 0;

That equation does not seem to be amenable to simplification. The special case of , however,
is much nicer:
> 
eval(eq_tmp1, {L[1]=L, L[2]=L}):
eq_tmp2 := simplify(%*4);

Let :
> 
eq_tmp3 := algsubs(L*mu=lambda, eq_tmp2);

That expression grows like , so we divide through by that to obtain
a betterbehaved equation
> 
eq := eq_tmp3/cosh(lambda)^2;

> 
plot(lhs(eq), lambda=0..2*Pi);

Here are the first three roots:
> 
lambda__1, lambda__2, lambda__3 :=
fsolve(eq, lambda=1e3..2*Pi, maxsols=3);

> 
params := { L[1]=1, L[2]=1 };

> 
mu__1, mu__2, mu__3 := (lambda__1, lambda__2, lambda__3) /~ eval(L[1], params);

> 
plots:display([
plot_beam(M, mu__1, params),
plot_beam(M, mu__2, params),
plot_beam(M, mu__3, params)],
color=["red","Green","blue"],
legend=[mode1, mode2, mode3]);



A dualspan clampedpinnedfree cantilever beam


We have a twospan beam of span lengths and , with the left end of the
first span clamped, the right end of the second span free, and the interface
between the spans on a pinned support. This is different from the previous
case only in the nature of the left boundary condition.
> 
M := build_matrix(
M_left_clamped,
M_right_free(mu,L[2]),
[ M_interface(mu,L[1])] );

The characteristic equation:
> 
Determinant(M);
eq_tmp1 := simplify(value(%)) = 0;

That equation does not seem to be amenable to simplification. The special case of , however,
is much nicer:
> 
eval(eq_tmp1, {L[1]=L, L[2]=L}):
eq_tmp2 := simplify(%*4);

Let :
> 
eq_tmp3 := algsubs(L*mu=lambda, eq_tmp2);

That expression grows like , so we divide through by that to obtain
a betterbehaved equation
> 
eq := eq_tmp3/cosh(lambda)^2;

> 
plot(lhs(eq), lambda=0..2*Pi);

Here are the first three roots:
> 
lambda__1, lambda__2, lambda__3 :=
fsolve(eq, lambda=1e3..2*Pi, maxsols=3);

> 
params := { L[1]=1, L[2]=1 };

> 
mu__1, mu__2, mu__3 := (lambda__1, lambda__2, lambda__3) /~ eval(L[1], params);

> 
plots:display([
plot_beam(M, mu__1, params),
plot_beam(M, mu__2, params),
plot_beam(M, mu__3, params)],
color=["red","Green","blue"],
legend=[mode1, mode2, mode3]);



A triplespan freepinnedpinnedfree beam


We have a triplespan beam with span lengths of . The beam is supported
on two internal pinned supports. The extreme ends of the beam are free.
The graphs of the first three modes agree with those
in Figure 3.22 on page 70 of the 2007 article of
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö, and Lars Håkansson.
In the caption of Figure 2.12 on page 28 the span lengths are given
as 3.5, 5.0, 21.5.
> 
interface(rtablesize=12):

> 
M := build_matrix(
M_left_free,
M_right_free(mu,L[3]),
[ M_interface(mu,L[1]), M_interface(mu,L[2])] );

> 
params := { L[1]=3.5, L[2]=5.0, L[3]=21.5 };

The characteristic equation
> 
simplify(Determinant(M)):
value(%):
eq := simplify(eval(%, params));

That graphs grows much too fast to be useful. We moderate it by dividing through
the fastest growing cosh term:
> 
plot(eq/cosh(21.5*mu), mu=0..0.4);

Here are the first three roots:
> 
mu__1, mu__2, mu__3 := fsolve(eq, mu=1e3..0.4, maxsols=3);

> 
plots:display([
plot_beam(M, mu__1, params),
plot_beam(M, mu__2, params),
plot_beam(M, mu__3, params)],
color=["red","Green","blue"],
legend=[mode1, mode2, mode3]);



A triplespan freespringspringfree beam


We have a triplespan beam with span lengths of . The beam is supported
on two internal springy supports. The extreme ends of the beam are free.
The numerical data is from the worksheet posted on July 29, 2020 at
https://www.mapleprimes.com/questions/230085ElasticfoundationMultispanEulerBernoulliBeamthreespan#comment271586
The problem is pretty much the same as the one in the previous section, but the
pinned supports have been replaced by spring supports.
This section's calculations require a little more precision than
Maple's default of 10 digits:
> 
interface(rtablesize=12):

> 
M := build_matrix(M_left_free, M_right_free(mu,L[3]),
[ M_interface(mu, L[1], 'Kd'=kd/(E*I), 'Kt'=kt/(E*I)),
M_interface(mu, L[2], 'Kd'=kd/(E*I), 'Kt'=kt/(E*I)) ]);

Calculate the determinant of . The result is quite large, so we terminate the command
with a colon so that not to have to look at the result. If we bothered to peek, however, we
will see that the determinant has a factor of . But that quite obvious by looking at the
entries of the matrix shown above. Two of its rows have in them and another two have
. When multiplied, they produce the overall factor of .
Here are the parameters that the determinant depends on:
> 
indets(DET, name); # the parameters that make up M

So we provide values for those parameters:
> 
params := {
L[1]=3.5, L[2]=5.0, L[3]=21.5,
kd=4.881e9, kt=1.422e4,
E = 2.05e11, I = 1.1385e7 };

Here is the characteristic equation. We multiply it by to remove the singularity at
> 
mu^8 * value(DET):
eq := eval(%, params):

We can't see anything useful in that graph. Let's limit the vertical range:
> 
plot(eq, mu=0..0.6, view=1e8..1e8);

> 
mu__1, mu__2, mu__3 := fsolve(eq, mu=1e3..0.6, maxsols=3);

> 
plots:display([
plot_beam(M, mu__1, params),
plot_beam(M, mu__2, params),
plot_beam(M, mu__3, params)],
color=["red","Green","blue"],
legend=[mode1, mode2, mode3]);

> 
Digits := 10; # restore the default


