LAB 1 Dynamic Equations of a Spring-Mass-Damper System

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LAB 1: Dynamic Equations of a Spring-Mass-Damper System
Objectives : Physical setup
Building the model with Simulink
Analysis and explanation.
Physical setup
Newton's laws of motion form the basis for analyzing mechanical systems. Newton’s
second law, Equation (1), states that the sum of the forces acting on a body equals
the product of its mass and acceleration. Newton's third law, for our purposes, states
that if two bodies are in contact, then they experience the same magnitude contact
force, just acting in opposite directions.
(1)
The free-body diagram for this system is shown below. The spring force is
proportional to the displacement of the mass, , and the viscous damping force is
proportional to the velocity of the mass, . Both forces oppose the motion of

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the mass and are, therefore, shown in the negative X-direction. Note also
that corresponds to the position of the mass when the spring is upstretched.
Now we proceed by summing the forces and applying Newton’s second law,
Equation (1), in each direction. In this case, there are no forces acting in the Y-
direction; however, in the X-direction we have:
(2)
This equation, known as the governing equation, completely characterizes the
dynamic state of the system. Later, we will see how to use this to calculate the
response of the system to any external input , as well as to analyze system
properties such as stability and performance.
Building the model with Simulink
To build the simulation model, open Simulink and open a new model window. Then
follow the steps listed below.
1. Insert an Integrator block from the Simulink/Continuous library and draw lines
to and from its input and output terminals.
2. Label the input line "Xddot" and the output line "Xdot". To add such a label,
double-click in the empty space just below the line.
3. Insert another Integrator block above the previous one and draw lines to and
from its input and output terminals.
4. Etc…
For the Second Order Differential Equation (Single Degree of Freedom System-
SDOF) model the initial conditions for both integrator blocks should be set to zero. As
for the gain blocks, use the following values m = 2, b = 2, and k = 4.

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Where (xddot) is the acceleration of the mass m, (xdot) is the velocity, x is the
displacement, f(t) is the forcing function (input force), b is the damping coefficient,
and k is the spring constant.
To model this system we start by looking at the terms in the bracket. There are three
input signal lines: f(t), xdot, and x. We also notice that both xdot and x are multiplied
by constants.
Analysis:
1. Provide an analytic explanation of the giving formula 2?
2. Provide the final Simulink design of the Spring-Mass-Damper System?
3. Run the simulation using the given parameter?
4. Draw the function: Force= f (time) using an appropriate scale?