
In this lecture we'll repeat the validation exercise we did for the forced vibration response; we'll use direct integration to confirm the modal superposition response calculated in the last lecture. Again, we'll use `ODEINT` from `SciPy`.

```python
# Use ODEINT to solve the differential equations
from scipy.integrate import odeint
```

The system of equations remains the same. We simply need to alter the initial conditions and set the magnitude of the forcing term to zero.

```python
def systemEquations(inits,t,params,):
"""
Defines the differential equations for the coupled system

var: vector of the state variables: var = [u1,y1,u2,y2,u3,y3]
    u1 = mass 1 position
    y1 = mass 1 velociity
    u2 = mass 2 position
    y2 = mass 2 velocity
    u3 = mass 3 position
    y3 = mass 3 velocity
t: time
params:  vector of parameters: params = [m1,m2,m3,k1,k2,k3,c1,c2,c3]
"""

# Unpack position and velocity variables and the system parameters
u1, y1, u2, y2, u3, y3 = inits # Position and velocity variables
m1, m2, m3, k1, k2, k3, c1, c2, c3, P, w = params # System parameters

#Define the force vector
F = F3*np.sin(w*t)

# Define a system of first order differential equations
#(introduce dummy variables y1 and y2 for this)
f = [y1,
     ( -y1*(c1+c2) + y2*c2 -(k1+k2)*u1 + k2*u2)/m1,
     y2,
     (y1*c2 - y2*(c2+c3) + y3*c3 + u1*k2 - u2*(k2+k3) + u3*k3)/m2,
    y3,
    (F + y2*c3 - y3*c3 + u2*k3 - u3*k3)/m3]

return f
```

Defining the initial conditions next.

```python
# Initial conditions
# x1, x2 and x3 are the initial displacements;
# y1, y2 and y3 are the initial velocities
x1 = 0.012 #initial displacement of mass 1
y1 = 0.0 #initial velocity of mass 1
x2 = 0.019 #initial displacement of mass 2
y2 = 0.0 #initial velocity of mass 2
x3 = 0.021 #initial displacement of mass 2
y3 = 0.0 #initial velocity of mass 2

#Set the damping to zero
c1 = 0
c2 = 0
c3 = 0

#Set the forcing term magnitude and frequency to zero
F3 = 0
w=0

# Pack up the parameters and initial conditions:
params = [m1, m2, m3, k1, k2, k3, c1, c2, c3, F3, w]
inits = [x1, y1, x2, y2, x3, y3]

# Call the ODE solver.
sol = odeint(systemEquations, inits, t, args=(params,))

#Assign displacements to new variables
U1_direct = sol[:,0] #Displacement of mass 1
U2_direct = sol[:,2] #Displacement of mass 2
U3_direct = sol[:,4] #Displacement of mass 3
```

### Plot the response obtained by direct integration

```python
fig, axes = plt.subplots(figsize=(12,6),nrows=1,ncols=1)
axes.plot(t,U1_direct,'-',label='mass 1 (first floor)')
axes.plot(t,U2_direct,'-',label='mass 2 (second floor)')
axes.plot(t,U3_direct,'-',label='mass 3 (third floor)')
axes.set_xlabel('Time (sec)')
axes.set_ylabel('Displacement (m)')
axes.set_title('Structural response')
axes.set_xlim([0,20])
axes.grid()
axes.legend(loc='lower right')
plt.show()
```

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/compare-modal-superposition-with-direct-integration-free-vibration/img1.jpg"
	alt="Free vibration structural response of masses 1, 2 and 3 (top to bottom) obtained from direct integration. | EngineeringSkills.com"
	width="1000"
	height="519"
	caption="Fig 1. Free vibration structural response of masses 1, 2 and 3 (top to bottom) obtained from direct integration."
/>

### Compare the modal response with the direct integration response

```python
fig, axes = plt.subplots(figsize=(12,14),nrows=3,ncols=1)
axes[0].plot(t,u1,'-',label='Modal superposition')
axes[0].plot(t,U1_direct,'--',label='Direct integration')
axes[0].set_xlim([0,tmax])
axes[0].set_ylabel('Structural Displacement (m)')
axes[0].set_title('Mass 1')
axes[0].grid()
axes[0].legend(loc='lower right')

axes[1].plot(t,u2,'-',label='Modal superposition')
axes[1].plot(t,U2_direct,'--',label='Direct integration')
axes[1].set_xlim([0,tmax])
axes[1].set_ylabel('Structural Displacement (m)')
axes[1].set_title('Mass 2')
axes[1].grid()
axes[1].legend(loc='lower right')

axes[2].plot(t,u3,'-',label='Modal superposition')
axes[2].plot(t,U3_direct,'--',label='Direct integration')
axes[2].set_xlim([0,tmax])
axes[2].set_xlabel('Time (sec)')
axes[2].set_ylabel('Structural Displacement (m)')
axes[2].set_title('Mass 3')
axes[2].grid()
axes[2].legend(loc='lower right')

plt.show()
```

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/compare-modal-superposition-with-direct-integration-free-vibration/img2.jpg"
	alt="Comparison of free vibration structural response of masses 1, 2 and 3 (top to bottom) obtained from direct integration and modal superposition. | EngineeringSkills.com"
	width="1000"
	height="1093"
	caption="Fig 2. Comparison of free vibration structural response of masses 1, 2 and 3 (top to bottom) obtained from direct integration and modal superposition."
/>

As expected, both response calculations agree. Now that we've reached the end of this section on modal analysis you should have a clear understanding of how to perform a modal analysis and why you would opt to perform such an analysis over simply plugging your equations into a numerical solver such as <code>ODEINT</code>. In the next section we'll address damping and see how we can implement damping within our modal analyses.


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