As I said in Introduction of Engineering Math page, Mathematics is a kind of language. In any kind of language learning, the first step is to learn the vacabularies/words and the rules to combine those words into a sentence. The intention of this section is to provide you a handy dictionary of words for a language called 'Differential Equation'. But don't try to blindly read through this section from the beginning to the ends. You may have not seen any person who is trying to bindly remember every words in a dictionary from the first page to the last page. Normally you would try to speak a simple sentence or you would hear somebody say some sentences and then you would look into a dictionary when you come accross any word that you don't know. You can use this section in the same manner. See 'Modeling Example' section first (these are kinds of sentences written in differential equation and get back to this section when you have difficulties in understanding how the sentence is derived)

I will start out with very short list of dictionary but I will keep updading this section to make it thicker and thicker.

• What are the mechanical building blocks ?
• What are the electrical building blocks ?
• How do the mechanical and electrical blocks map to each other ?

What are the mechanical building blocks ?

Let's start with the mechanical blocks. Each block links one input, the applied force or torque, to one output, the displacement or angle. The blocks differ only in which derivative of the output appears in the equation. A spring uses the output itself, a damper uses its first derivative and a mass uses its second derivative.

Mechanical Building Block : Spring

The spring is the simplest block, because its force depends only on the present displacement. The applied force F(t) is the input, and the displacement x(t) is the output. The spring constant k links the two, so doubling the force doubles the stretch.

Mechanical Building Block : Damper

A damper resists motion rather than position. Its force depends on how fast the rod moves, so the equation below carries the first derivative dx/dt. The damping coefficient k_d sets the force per unit velocity. A damper that does not move carries no force at all, whatever its position.

Mechanical Building Block : Mass

A mass needs no force to stay where it is or to keep a constant speed. It needs force only to change its speed, and that is Newton's law F = ma. So the mass block brings the second derivative d²x/dt² into the model.

Mechanical Building Block : Spring-Mass

Here two blocks share one displacement x(t), so their forces add. The applied force must stretch the spring and accelerate the mass at the same time. The result is a second order equation. With no applied force, its motion is an oscillation at the angular frequency ω = √(k/M).

Mechanical Building Block : Spring-Damper

In this block the spring and the damper sit side by side between the input point and the wall, so both see the same x(t). Their forces add, and the result is a first order equation. With a constant force F, x(t) rises toward F/k with the time constant k_d/k.

Mechanical Building Block : Spring-Mass-Damper

This block adds all three forces, so the equation below is the standard second order model of a vibrating system. Each term keeps the label it had as a single block. You will meet this equation again in almost every vibration problem.

Note one detail of the drawing. It places the damper between the mass and the input force. But the equation treats the damper as acting on the velocity of the mass itself. That is the case when the damper is fixed between the mass and the wall. For the damper in series as drawn, the damping force would depend on the difference between the velocity of the input end and the velocity of the mass. The equation is the standard model, and it matches the parallel layout. The damping ratio ζ = k_d/(2√(kM)) then tells you how the free motion looks. It oscillates while ζ is below 1, and it settles without oscillation when ζ is 1 or more.

Mechanical Building Block : Rotational Bar

A rotating block follows the same pattern as a moving one, with the torque T(t) in place of force and the angle θ(t) in place of displacement. A bar twisted at one end acts as a torsional spring. Its torque grows in proportion to the twist angle, and k is the torsional stiffness.

Mechanical Building Block : Rotational Damper

A paddle that turns in a fluid resists the rate of rotation rather than the angle. This is the rotational version of the damper. Its torque is k_d times the angular velocity dθ/dt, so it brings the first derivative of the angle into the model.

Mechanical Building Block : Rotational Mass - Rotational Inertia

A disk on a shaft needs torque only to change its rate of rotation. Its moment of inertia I plays the role that the mass M plays in straight-line motion. So the equation below is Newton's law for rotation, with the second derivative of the angle.

• Spring, damper and mass differ by one derivative each : the spring force follows x, the damper force follows dx/dt and the mass force follows d²x/dt².
• Blocks that share one displacement add their forces : every combined block on this page is the sum of the single blocks it contains.
• Rotation reuses the same three blocks : torque replaces force, angle replaces displacement and moment of inertia replaces mass.

What are the electrical building blocks ?

The electrical blocks work the same way, with the applied voltage V(t) as the input and the current i(t) as the output. Again the three blocks differ by what they do to the output. The resistor multiplies the current by a constant, the inductor differentiates it and the capacitor integrates it.

Electrical Building Block - Reister

The resistor is the electrical block with no memory. Its voltage depends only on the present current, through Ohm's law, so it brings no derivative into the model. In the analogy at the end of this page, it plays the role of the damper.

Electrical Building Block - Inductor

An inductor resists a change of current. Its voltage is proportional to di/dt, so a steady current produces no voltage across it. The constant L is the inductance, measured in henry.

Electrical Building Block - Capacitor

A capacitor stores charge, and its voltage follows the charge it has collected. The charge is the integral of the current, so the equation below carries an integral. Differentiating both sides gives the same law in derivative form, i(t) = C dV/dt.

When you connect these blocks in one loop, KVL adds their voltages, in the same way that the forces added in the combined mechanical blocks. A loop with one inductor or one capacitor gives a first order equation. A loop with both gives a second order equation.

• The resistor has no memory : its voltage follows the present current, so it adds no order to the equation.
• The inductor and the capacitor store energy : each one brings a derivative or an integral, and each adds one order.
• The capacitor law has two forms : the integral form in the picture and the derivative form i = C dV/dt say the same thing.

How do the mechanical and electrical blocks map to each other ?

You may have noticed that the two sets of equations above have the same shape. Let's line them up, because the match lets you reuse the solution of one system for the other.

Write the capacitor law with the charge q, where i = dq/dt. A series loop of L, R and C driven by V(t) then gives V = L d²q/dt² + R dq/dt + q/C. Compare it with the spring-mass-damper equation F = M d²x/dt² + k_d dx/dt + kx. The two equations match term by term if you read force as voltage and displacement as charge. This pairing is called the force-voltage analogy, and the table below lists it for all the blocks on this page.

The natural frequency follows the same pairing. It is √(k/M) for the spring-mass block and 1/√(LC) for the loop, because k maps to 1/C and M maps to L. So a result you work out for one system carries over to the other by renaming the constants.

• One equation serves three kinds of system : the translational, rotational and electrical second order models differ only in the names of their constants.
• The spring maps to the inverse of the capacitance : a stiff spring corresponds to a small capacitor, which is the one pairing that is easy to get backwards.