Mathematical Model of Mechanical Systems
Mechanical Systems can be classified into two types based on their type of motion:
- Translational Systems – Linear motion
- Rotational Systems – Circular motion
Translational Systems
Translational Systems are characterized by movement in straight lines and primarily consist of three basic elements – masses, springs and dampers. Consider the following translational mechanical system.
Translation Mechanical System
A mass ‘M’ is tethered to a fixed rigid support via a spring (with spring constant ‘K’), and the friction between the mass ‘M’ and the fixed surface is indicated by a damper with viscous damping coefficient ‘B’. An external force F(t) is being applied to this mass, causing a displacement x(t) in the direction of the applied force. Thus, the free body diagram of the Mass block can be drawn as follows.
Mass Block Free Body Diagram
Now, according to Newtons second law, the sum of all external forces applied on a body is directly related to the acceleration it undergoes in the same direction, and inversely proportional to its mass.
∑ External Forces = Mass ✖ Acceleration
bringing the right hand side to the left,
∑ F – ma = 0
and then considering the ‘ma’ term to be a force itself, we are left with D’Alembert’s Law
∑ F = 0
essentially implying that the algebraic sum of all the forces acting on a mechanical system is zero. In other words, the sum of all applied forces is equal to the sum of all opposing forces.
Going back to the considered system, and applying this law,
Externally applied force = Inertial force + Frictional force + Restoring force of Spring
[Tex]F(t) [/Tex] = [Tex]F_m [/Tex] + [Tex]F_{\text{friction}} [/Tex] + [Tex]F{\text{spring}} [/Tex]
F(t) = Ma(t) + Bv(t) + Kx(t)
[Tex]F(t) = M \frac{d^2x(t)}{dt^2} + B \frac{dx(t)}{dt} + kx(t) [/Tex]
Taking the Laplace Transform of this equation (assuming initial conditions to be zero), we get the s-domain equation modeling a translational mechanical system
[Tex]F(s) = Ms^2X(s) + BsX(s) + KX(s) [/Tex] ⇒ Eq. 1
Force Voltage Analogy
In this Article, We will be going to Know what is Force Voltage Analogy, We will go through the Mathematical Model of the Mechanical System Which is Classified into Two types Translational and Rotational Systems. Then we go through the Mathematical Model of the Electrical System, Then we go through the Force Voltage Analogy, At last, we will Conclude our Article With its Applications and Some FAQs.
Table of Content
- What is the Force Voltage Analogy?
- Mathematical Model of Mechanical Systems
- Mathematical Model of Electrical Systems
- Force-Voltage Analogy
- Translational Mechanical to Electrical System Conversion Example
Contact Us