Mechanical Component Modeling
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## 1 Introduction to Modeling of Mechanical Systems

### 1.1 Some useful equations

Figure 1: Typical Single Mass, Spring, Damper system

As anyone who has gone through Mechanical engineering knows, many real world systems can be approximated by combining spring, mass, and damper elements into an idealized model of the real world system. Tables 1 and 2 present the equations for these elements.

Table 1: Linear Mechanical Elements - Free Body Diagram Equations
Description Free Body Diagram Equation
Damper $LaTeX: F=\plusmn B\left(\nu_1 \plusmn \nu_2\right)$
Spring $LaTeX: F=\plusmn k\left(x_1 \plusmn x_2\right)$
Mass $LaTeX: \sum F=m\ddot{x}=m\frac{d\nu}{dt}$
Notes:
1. Typically the force of damper is stated as being $LaTeX: F=-B\nu=-B\dot{x}$. This is always a relative velocity between the 2 bodies connected by the damper. In this table that is explicit and instead of using the implicit relative velocity $LaTeX: \nu$ the table uses $LaTeX: \nu_1 \plusmn \nu_2$. The equivalent is true for the spring position.
2. Dampers are also known by the names dashpot or linear friction.
3. $LaTeX: \sum F$ is the sum of all forces (Newton's 2nd law).
 TODO Add Rotational equivalent to tables

Table 2: Linear Mechanical Elements - Energy
Description Energy
Power dissipation in Damper $LaTeX: P=F\nu=F^2\frac{1}{B}=\nu^2B$
Energy stored in Spring $LaTeX: E=\frac{1}{2}k\left(\Delta x\right)^2$ or $LaTeX: E=\frac{1}{2} \frac{1}{k} f^2$
Energy stored in Mass $LaTeX: E=\frac{1}{2} m\nu^2$

## 2 Derivation of Differential Equations

Differential equations become transfer functions through the Laplace transform. State space equation are formed by directly from the differential equations (sometimes this requires some algebraic rearranging).

This is hard part and it just takes practice. Here is procedure to help:

1. Identify all elements which can move indepently. Typically these elements have mass or inertia.
2. Determine the type of motion for each element. Examples are
• 1D translation (such as sliding along a frictionless surface)
• 1D rotation (such as a top)
• 2D translation or planar motion (again sliding on a table but in doing so in both x and y axes)
• 3D translation
• 3D rotation (such as a spacecraft's attitude)
• 6 Degree of Freedom (DOF) rigid body motion (such a spacecraft's attitude and position as it goes around it's orbit)
3. Identify force/torque elements. Examples are
• Springs and dampers
• Motors
• Stiffness of a beam with 1 end fixed and the other rotating or twisting
4. Draw a Free Body Diagram that includes
• Each element capable of independent motion
• Interconnections - springs, damper, rigid elements
• External forces
• Geometry relating each independent element to the forces that act on it
• Dimension, coordinate systems, angles, etc. for each independent element
• Sign conventions

### 2.2 Use Newton's 2nd Law

Newton's 2nd Law is the basis of almost all statics and dynamical system modeling. For reference it is below.

 $LaTeX: \sum F=ma=m\ddot{x}$ or $LaTeX: \sum \Tau=J\ddot{\theta}$

where

• $LaTeX: \sum F$ is the sum of all forces and $LaTeX: \sum \Tau$ is the sum of all torques,
• $LaTeX: m$ is the mass and $LaTeX: J$ is the inertia,
• $LaTeX: x$ is the translational position or displacement and $LaTeX: \theta$ is the angular position, and
• $LaTeX: a=\ddot{x}$ is the acceleration and $LaTeX: \ddot{\theta}$ is the angular acceleration.

With a little practice this part can become a matter of mechanical manipulation of the idealized model. In other words, once you become proficient in creating the idealized model this part is essentially a matter of remembering all the implications of Newton's 2nd Law and exercising your geometry and algebra skills.

1. Determine and write down the position, angle, or both of each independent element
2. Using the above, deduce (via geometry, algebra, and differentiation) the following
• deflection of all springs
• rate of deflection of all dampers
• inertial linear acceleration of all masses
• inertial angular acceleration of all rotary inertias
3. Write down all sign conventions (should already be done but just in case...)
4. Use Newton's 2nd Law to write differential equations for each free body (mass, connections, etc.)
• There should be 1 equation for each degree of freedom (i.e., 1 equation for each unknown); if you have fewer equations than degrees of freedom then you will not be able to find a unique solution but rather a solution set
5. Use Table 1 and Table 2 to determine the forces applied by springs and dampers; substitute these equations (constitutive laws) into the equation set you created in the previous step using Newton's 2nd Law