Name: transfer functions in mechanical motion
Uploaded: 2024-09-26T13:17:18.000Z
Duration: 1 min 22 s
Channel: JoVE

K12 Standards

Texas Essential Knowledge and Skills

High School

Physics

PHYS.1

IPC.1.G

Mechanical Systems

In mechanical systems, springs, and masses are akin to the roles of inductors and capacitors in electrical networks, with the energy-dissipating function of a viscous damper corresponding to that of electrical resistance.
The forces acting on the mass include an applied force moving in the same direction and the forces from the spring, viscous damper, and acceleration acting against it.
Like the RLC network, translational mechanical systems are defined by a unique differential equation formulated by applying Newton&#39;s law which mandates that the sum of all forces acting on the mass must equal zero.
Further, the Laplace transform is used on the equation under zero initial conditions. This expression, when simplified, yields the transfer function.
The components in rotational mechanical systems mirror those of translational systems but experience rotation.
These are managed similarly to translational ones, with torque substituting force, angular displacement replacing translational displacement, and inertia replacing the mass.
In the second-order differential equation for the rotational system, the Laplace transform is applied and then further simplified to yield the transfer function.

mechanical systems

Mechanical systems are analogous to to electrical networks where springs and masses play similar roles to inductors and capacitors, respectively. A viscous damper in mechanical systems functions similarly to a resistor in electrical networks, dissipating energy. The forces acting on a mass in such systems include an applied force in the direction of motion, counteracted by forces from the spring, a viscous damper, and the mass&#39;s acceleration. This interplay of forces is mathematically described using Newton&#39;s second law, which states that the sum of all forces acting on the mass must be zero.
In translational mechanical systems, the behavior is captured by a unique differential equation derived from Newton&#39;s law. This equation accounts for all the forces acting on the mass. To solve the system analytically, the Laplace transform is applied to this differential equation under zero initial conditions. The Laplace transform, a powerful mathematical tool, converts the time-domain differential equation into an algebraic equation in the Laplace domain. Simplifying this equation yields the system&#39;s transfer function, a crucial concept that relates the output response to the input force in the frequency domain. The transfer function is essential for analyzing system stability and dynamics.
Rotational mechanical systems parallel translational systems but involve rotational motion. In these systems, torque replaces force, angular displacement substitutes for translational displacement, and rotational inertia takes the place of mass. The analogous differential equation for a rotational system, derived similarly using Newton&#39;s second law for rotation, describes the dynamics of rotational motion. By applying the Laplace transform to this second-order differential equation, and simplifying, the transfer function for the rotational system is obtained. This function provides insights into the rotational system&#39;s behavior, similar to how the transfer function in translational systems aids in understanding linear motion dynamics.

In mechanical systems, springs, and masses are akin to the roles of inductors and capacitors in electrical networks, with the energy-dissipating function ...

Education

JoVE Core

Electrical Engineering

Unit 1

Modeling in Time and Frequency Domain

KEY TERMS AND CONCEPTS

transfer functions in mechanical motion

Mechanical motion can be modeled with springs, masses, and viscous dampers. In these models, springs behave like inductors and masses act like capacitors in an electrical network. A viscous damper works like a resistor because it removes energy from the system.

A force acts on the mass in the direction of motion. It is balanced by the spring force, the damping force, and the force from the mass’s acceleration. Newton’s second law is used to describe this balance, and the sum of the forces on the mass must be zero.

For translational mechanical systems, this force balance leads to a second-order differential equation. The equation captures the motion of the mass under the applied force. To solve it analytically, the Laplace transform is applied under zero initial conditions. This changes the time-domain equation into an algebraic equation in the Laplace domain.

After simplification, the transfer function is obtained. A transfer function relates the output response to the input force in the frequency domain. It is a key tool for studying stability and system dynamics.

Rotational mechanical systems follow the same overall idea, but the motion is rotational. In this case, torque replaces force, angular displacement replaces translational displacement, and rotational inertia replaces mass. A similar second-order equation comes from Newton’s second law for rotation.

Applying the Laplace transform to the rotational equation gives the rotational transfer function. This function helps describe how the rotational system behaves. It serves the same purpose for rotational motion that the translational transfer function serves for linear motion.

Transfer Functions in Mechanical Motion

transfer function in control systems

Transfer Function from Input to Output

The transfer function describes how a linear time-invariant, or LTI, system maps an input to an output in the frequency domain. It gives a compact way to study system behavior without working only in the time domain. It also connects the differential equations of the system to a form that is easier to handle mathematically. For an nth-order LTI system, the behavior is written as a differential equation with constant coefficients. In that equation, r(t) is the input and c(t) is the output. The...

electrical systems

RLC Circuit Transfer Functions

RLC circuits use resistors, capacitors, and inductors to show how electrical signals change over time. In circuit analysis, these passive linear components are arranged so the link between an input and an output can be described with a transfer function. For this topic, the output is the voltage across the capacitor, compared with the input voltage. A series RLC circuit can be analyzed with Kirchhoff's Voltage Law, or KVL. KVL says that the sum of all voltages around a closed loop is zero.

Mechanical motion can be modeled with springs, masses, and viscous dampers. In these models, springs behave like inductors and masses act like capacitors in an electrical network. A viscous damper works like a resistor because it removes energy from the system. A force acts on the mass in the direction of motion. It is balanced by the spring force, the damping force, and the force from the mass’s acceleration. Newton’s second law is used to describe this balance, and the sum of the forces on...

electro-mechanical systems

DC Motor Modeling in Electromechanical Systems

DC motor modeling is a key part of electromechanical systems. These systems combine electrical parts and mechanical parts to produce a useful output. A DC motor is one common example because it turns electrical energy into mechanical motion. It is used in many places, from simple fans to complex robotic mechanisms. Inside the DC motor, the armature is a rotating circuit placed in a magnetic field. When current flows through the armature, it experiences a force from the magnetic field. That...

linear approximation in frequency domain

Linearizing Nonlinear RL Circuits

Linearizing nonlinear RL circuits helps turn a hard circuit problem into a simpler linear one. Linear systems follow superposition, which means the total response equals the sum of the responses to separate inputs. They also follow homogeneity, which means scaling the input scales the output by the same amount. Nonlinear systems do not automatically have those properties. For small changes around an operating point, though, a nonlinear system can often be treated as linear. This is done with a...

state space representation

RLC Circuit State Equations

RLC circuit state equations show how a circuit changes over time using state variables. This time-domain method is useful for linear, time-invariant systems, but it also helps when frequency-domain analysis is not enough. It is especially useful for nonlinear systems, time-varying systems, and multiple-input multiple-output systems. The state-space approach turns an nth-order system into a set of simultaneous first-order differential equations called state equations. For a common second-order...

transfer function to state space

Phase-Variable State Equations from Transfer Functions

Phase-variable state equations show how a transfer function can be rewritten for computer simulation. In state-space form, the system is described with state variables, which are the output and its first several derivatives. This makes it easier to model a physical system on a digital computer. The starting point is an nth-order linear differential equation with constant coefficients. Systems like an RLC circuit can be written this way. To build the state equations, the transfer function is...

state space to transfer function

Finding a Transfer Function from State Space

Finding a transfer function from state-space form is a key step in system analysis. It changes a time-domain model into a frequency-domain form. That makes it easier to study and design control systems. The process starts with the state-space equations. These include the state equation and the output equation. In these equations, x(t) is the state vector, u(t) is the input vector, y(t) is the output vector, and A, B, C, and D are the matrices that define the system dynamics. Next, the Laplace...

linear approximation in time domain

Linearizing Nonlinear State-Space Models

Linearizing nonlinear state-space models helps describe systems that change with time or operating conditions. State-space representation is often effective for this kind of modeling. It is especially useful for systems such as a simple pendulum or a translational mechanical system with nonlinear springs. A simple pendulum can be modeled by summing torques around the pivot point. In this case, the mass is evenly distributed along the pendulum, and the center of mass is located at half its...

key terms and concepts

Learn how springs, masses, and dampers are modeled with Newton’s law and the Laplace transform to find transfer functions in translational and rotational motion.

Transfer Functions in Mechanical Motion | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Chemistry and Physics

IPC.1

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PHYS.1.H

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Transfer Functions in Mechanical Motion

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