
In this lecture we simply want to compare the vibration of our structure, obtained using modal superposition, with the result obtained from direct integration. We've solved this system using direct integration already, by implementing a central difference algorithm and also using `ODEINT` from `SciPy`. We'll repeat the solution here using `ODEINT`, except this time we'll set the damping to zero to match our modal superposition analysis. We can start by copying in the function we wrote in the last section that returns our system equations.

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

```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 = 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
```

Next we define our initial conditions and pass those along with the other system parameters and our previously defined function into `ODEINT`.

```python
# Initial conditions
# x1, x2 and x3 are the initial displacements;
# y1, y2 and y3 are the initial velocities
x1 = 0.0 #initial displacement of mass 1
y1 = 0.0 #initial velocity of mass 1
x2 = 0.0 #initial displacement of mass 2
y2 = 0.0 #initial velocity of mass 2
x3 = 0.0 #initial displacement of mass 3
y3 = 0.0 #initial velocity of mass 4

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

# Pack up the parameters and initial conditions:
params = [m1, m2, m3, k1, k2, k3, c1, c2, c3]
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/comparing-modal-superposition-with-direct-integration-forced-response/img1.jpg"
	alt="Structural response of mass (floor) 1, 2 and 3 obtained by direct integration. | EngineeringSkills.com"
	width="1000"
	height="519"
	caption="Fig 1. Structural response of mass (floor) 1, 2 and 3 obtained by direct integration."
/>

The response obtained by solving the coupled equations of motion appears to match our modal superposition solution. But to properly compare, we can compare the response for each mode individually.

### 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/comparing-modal-superposition-with-direct-integration-forced-response/img2.jpg"
	alt="Comparison of responses obtained using modal superposition and direct integration for masses (floors) 1, 2 and 3 (top to bottom). | EngineeringSkills.com"
	width="1000"
	height="1093"
	caption="Fig 2. Comparison of responses obtained using modal superposition and direct integration for masses (floors) 1, 2 and 3 (top to bottom)."
/>

The plot above shows that both solution techniques provide the exact same response. We can expect the modal superposition of all $N$ modes of a NDoF system to yield the full response. Notice that when approaching this analysis from a modal perspective, we obtained far more insight into the dynamic behaviour of the system. We obtain valuable knowledge about what frequencies of loading the structure is particularly susceptible to. This is one of the key advantages of modal analysis.

Another advantage of modal analysis is that it offers us the ability to approximate the dynamic behaviour by only including in the superposition those modes that contribute the most to the response. This is particularly helpful when our dynamic model consistes of hundreds or thousands of discrete masses and therefore hundreds or thousands of modes! The response will be dominated by those modes with frequencies in the region of the forcing frequency or its integer multiples (harmonics).

As an example, consider the negligible difference in response when we compare a superposition of all modes with only the first mode, which dominates the response.

```python
# Approximate the response using only mode 1
u1m1 = Phi[0,0]*Y1
u2m1 = Phi[1,0]*Y1
u3m1 = Phi[2,0]*Y1
```

```python
fig, axes = plt.subplots(figsize=(12,14),nrows=3,ncols=1)
axes[0].plot(t,u1,'-',label='All modes')
axes[0].plot(t,u1m1,'--',label='Mode 1 only')
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='All modes')
axes[1].plot(t,u2m1,'--',label='Mode 1 only')
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='All modes')
axes[2].plot(t,u3m1,'--',label='Mode 1 only')
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/comparing-modal-superposition-with-direct-integration-forced-response/img3.jpg"
	alt="Comparison of structural responses obtained using contributions from all modes of vibration and just the first mode for masses (floors) 1, 2 and 3 (top to bottom). | EngineeringSkills.com"
	width="1000"
	height="1093"
	caption="Fig 3. Comparison of structural responses obtained using contributions from all modes of vibration and just the first mode for masses (floors) 1, 2 and 3 (top to bottom)."
/>

From the plots above it's pretty clear that a very sound approximation of the dynamic behaviour of the system is obtained from response in the first mode alone.


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