Laboratory # 3 Linear motionFlight DynamicsGroup 118/318

Abstract— The work shows the use of SIMULINK for modeling of mechanical systems. As an example a mechanical model of one mass-spring-damper is solved by this toolbox for the MATLAB/ Simulink environment and also by the use of differential equations.

Key Words: Mass Spring Damper System, Damping, Simulink Model

I. INTRODUCTION Mass-spring-damped models are used to study many

practical problems in engineering. It has a linear motion, which means its velocity, acceleration and position vary only in one axis. In addition to design and analysis of engineering and physical systems, these models provide insight into design of physical experiments.

II. BACKGROUND

A. Mathematical ModelFirst, One way to analyze the behavior of a dynamical

system is by means of a mathematical model. Such models are often described by (ordinary or partial) differential equations.

Newton’s second law leads to

m q̈+c ( q̇)+k q=0 .

Fig 1. Mass-Spring-Damper System

• q denotes the position of the mass (in a chosen coordinate system) and varies with time. q̇ and q̈ are the velocity and the acceleration.

• kq is the spring restoring force (assumed to satisfy Hooke’s law) and c c (q̇ ) is the friction force which can depend nonlinearly on the velocity.

B. InputsThe previous system is said to be autonomous since it is not exposed to external influences. Non-autonomous systems do have external inputs.

With an external force u(.) acting on the mass, we obtain

m q̈+c ( q̇)+k q=u(t )

Fig 2. Mass-Spring-Damper System Inputs

The external force u( .) typically varies with time. Depending on the circumstances it can be interpreted as follows:• If we are allowed to manipulateu( .)then it is called a control input.

• If u( .) is generated by nature and cannot be influenced/changed by us then it is called a disturbance input.

C. Outputs

Often not all variables that appear in a model are of interest. We choose outputs in order to describe those quantities that get focus.

If we are only interested in the po- sition of the mass, the output yis

m q̈+c ( q̇)+kq=u , y=q .

For some control input y (.) and along a system trajectory, the output y (.) will be a function of time. Interpretations of outputs:

Paola Elizabeth Sanchez Mireles 1606951Centro de Investigación e Innovación en Ingeniería Aeronáutica

Facultad de Ingeniería Mecánica y EléctricaUniversidad Autónoma de Nuevo León

• y is a variable that can be measured (through sensors).

• y indicate a variable which we would like to monitor in order to investigate/analyze the properties of the system (in simulation).

III. SIMULINK ANALISYS

The equation is defined and introduced to SIMULINK using integrators to simulate and operate the changes in the system. Then we establish the coefficients and also place a scope to

obtain the graphics.The diagram will be showed next:

Fig 3. Diagram blocks in SIMULINK

1. Simulate the system for f(t) = 0, x1(0)=1, x2(0)=-2, k=1 N/m, b=1 Ns/m and m=1Kg. f(t).

Once the simulation is done, we obtain:

Fig 4. Mass-spring-damper constant graph

The system stabilize nearly 50 seconds after the starting point.

2. Obtain the system performance for f(t) = 0, x1(0)=1, x2(0)=-2, k=1 N/m, b=1 Ns/m and m=1Kg. f(t) is the step function

Refreshing our system, we obtain:

Fig 5. Mass-spring-damper step graph

The stabilazing is almost the same and a change in the beginning is due to the step function. The frecuency of our system doesn’t change.

3. Obtain the system performance for f(t) = 0, x1(0)=1, x2(0)=-2, k=1 N/m, b=1 Ns/m and m=1Kg. f(t) is the ramp function

We obtain:

Fig 6. Mass-spring-damper ramp graph

Because of an increasing force the system does not stabilize at any point.

4. Now we change the parameters for f(t)=0, x1(0)=-3, x2(0)=2, k=10 N/m, b=5 Ns/m and m=2.5Kg.

We simulate our modified system with a constant, step and ramp function.

Fig 7. Mass-spring-damper constant graph

Fig 8. Mass-spring-damper step graph

Fig 4. Mass-spring-damper ramp graph

It is shown from the Scope1 that there is no oscillations By changing the parameters we can see that in the constant and step it does stabilize nearly at the same point in time but in the ramp function it doesn’t stabilize at all as the other example.

CONCLUSION

We conclude that a spring mass damper system, which is widely used in mechanical applications, can be well represented and simulated on a computer to reproduce real-life situations and accurately predict different conditions and outputs desired.

Thus it can be used to design systems, which have not been manufactured for testing.

REFERENCES

[1] http://www.mathworks.com/help/ident/gs/about-system- identification.html2015 - NCS Pearson.

[2] Shtessel, Yuri; Edwards, Christopher.; Fridman Leonid; Levant, Arie. “Sliding Mode Control and Observation”. 2014. XVII. P. 168.