so welcome to the lecture on multi degree
of freedom system so in this lecture we will
discuss the stiffness influence coefficient
so we discussed in the previous lecture the
flexibility influence coefficient and we saw
that if we know the flexibility influence
coefficient we can prepare the flexibility
matrix and by using that matrix we can write
the equation of motion
now we have similarly we have stiffness influence
coefficients because in equation of motion
of a multi degree of freedom system we have
m x double dot + k x equal to 0 for and multi
degree freedom system undamped system
so we can write here m x double dot + k x
equal to 0 so here k is stiffness matrix 
and m is the mass matrix so k is the stiffness
matrix and the elements of the stiffness matrix
or cover called the stiffness influence coefficients
so here k can be written as 
so we can see that the stiffness matrix that
is made from stiffness coefficients influence
coefficients and it is n by n square matrix
so its elements k 1 1 k 1 2 and k n n
so this we can find and we can make this matrix
and we can use directly in that equation so
we dont need to write the equation of motions
and prepare this matrix we can directly prepare
the stiffness matrix so here we concentrate
on one element k i j so here k i j what is
k i j?
so k i j is defined as the force or moment
required at any co-ordinate q i when a unit
displacement this displacement could - could
be either rectilinear or angular is applied
at co-ordinate j 
helding all other coordinates fixed or all
other degree of freedom fixed 
so here the stiffness influence coefficient
k i j is defined as a force or a moment so
it is force if it is the linear motion and
moment if it could be a angular motion
so what is the force that is required at a
coordinates i when unit load a unit displacement
is applied at j while all other degree of
freedom have zero or fixed so they have the
zero moment similarly k j i can be defined
as reverse like k j i is we are measuring
the force at j and we are giving a unit displacement
at i and again with this reciprocal theorem
this k j i is k i j
so therefore we will find this coefficients
for certain example so lets take one example
here and we try to find out this
so we have this two degree of freedom system
consisting of two masses m 1 and m 2 and to
stiffness elements k 1 and k 2 now we have
to find the stiffness influence coefficients
and so the stiffness matrix k so it will have
k 1 1 k 1 2 k 2 1 and k 2 2 because it is
two degree of freedom system so this matrix
will be square and 2 by 2 matrix and we will
have total 4 influence coefficient that we
have to find out
so 
now as k 1 1 or k 1 2 k 2 1 k 2 2 so k i j
as we define that we want to measure the force
that is developed as form of reaction when
we apply a unit load at j so here k i j so
the force at i when displacement unit unit
displacement at j so here now we give unit
displacement here at so this is unit displacement
so we are giving unit displacement to the
first mass in this direction and we are keeping
fixed other all the other degree of freedom
must be fixed
so this mass is fixed this is not moving now
when we give unit displacement we are going
to measure the forces so we are going to measure
here the force this and here the force so
this is k 1 1 because we are giving unit displacement
at 1 and measuring the force at 1 now we are
giving unit displacement at 1 and measuring
the force at 2 so this is k 2 1 so now we
will show the free body diagram for this system
so the free body diagram so this is my first
mass and this is the second mass so now this
first mass is having unit displacement so
when we are pulling so of course here the
force k 1 1 is here k 1 1 is acting on this
mass when we give unit displacement with this
spring will apply a force in this direction
opposite direction that is k 1 into displacement
and displacement is 1 so the total force is
k 1
now we are compressing this spring so this
spring will apply a force k 2 into displacement
so displacement is 1 so this force is k 2
now this mass it is fixed so the same there
is no any so here is a force k 2 1 on this
mass and then there is spring force that is
k 2 due to that spring and this will be opposite
to the this direction so now if we write the
equilibrium equations for these two free body
diagram two masses
so we will have for this first one k 1 1 minus
k 1 minus k 2 equal to 0 so this implies k
1 1 equal to k 1 + k 2 so this is 1 stiffness
influence coefficient that we have obtained
now for the second mass we have k 2 1 + k
2 equal to 0 and this implies that k 2 1 equal
to minus k 2 so we have got the second influence
coefficient so k 1 1 is the force that is
developed at 1 when we apply a unit displacement
at 1 while in k 2 1 is the force that is developed
at 2 when we apply a unit displacement at
1
now for second other two elements so we take
the case when we have this mass system 
now we are giving in this case a unit displacement
here so this is unit displacement to the second
mass - unit displacement 
and we are measuring the forces at both masses
so we here the force will be k 2 2 because
we are applying the displacement at 2 and
measuring the force at 2 and here it will
be k 1 2 we are applying a displacement at
2 and measuring the force at 1
now we make the free body diagram so the free
body diagram here we will make for these two
masses so for this one we will have here k
2 2 now we are pulling this spring so with
the unit displacement this will apply a force
k 2 into 1 so that is here and this spring
the same will apply the same force on this
mass then k 1 2 is another force that is working
here on this mass and this mass is fixed so
there will not - no be any stretching or compression
of the spring k 1
and therefore there will no be not be spring
force by this spring on this mass so we can
write the equations here for equilibrium equations
for these two masses 
so here we will have k 2 2 equal to k 2 and
from here we get k 1 2 + k 2 equal to 0 this
means k 1 2 equal to minus k 2
so we have obtained the all the four influence
coefficients stiffness influence coefficients
that is k 1 1 that is k 1 + k 2 k 2 1 that
is minus k 2 k 1 2 minus k 2 and k 2 2 equal
to k 2 so again from here we see that k 2
1 equal to k 1 2 equal to minus k 2 due to
the reciprocal theorem now we can write our
stiffness matrix 
so 
now we see that how we can find the stiffness
influence coefficients for multi degree of
freedom system and how the reciprocal theorem
according to reciprocal theorem we can omit
to calculate several terms and we can easily
find complete this stiffness matrix so this
is what i discussed for the multi degree of
freedom system that is undamped so we only
considered the mass and stiffness of the system
but we have also damped systems so we have
all the three elements that is mass stiffness
and damping elements so as for undamped system
we wrote the equations for multi degree of
freedom system
and we find represented in terms of matrices
so it was m x double dot + k x equal to 0
so this is this equation is for an undamped
spring mass system multi degree of freedom
system now if we have damping and we introduced
the damping in the system we have the c 1
c 2 c 3 terms then one more matrix will come
into this equation and we call it damping
matrix so we will we can write 
so this is equation of damped free vibration
of multi degree of freedom system
so we discussed the flexibility influence
coefficients we discussed the stiffness influence
coefficient now we have the damping matrix
and therefore it has the damping influence
coefficients so the damping matrix c 
so the damping matrix c can be written as
so this damping matrix c is also square matrix
of having n by n elements if it is a n degree
of freedom system and the elements of this
matrix are called the influence the damping
influence coefficients
so if we take this ci j so c i j it is defined
as the force or moment required at co-ordinate
i due to a unit velocity given at j co-ordinate
j while other degree of freedom have zero
velocity they are at rest so zero velocity
so similar to the stiffness coefficient we
can also calculate 
the coefficients of this damping matrix and
so we can find the damping influence coefficients
so i thank you for attending this lecture
and see you in the next lecture thank you
