This document summarizes research on using the transient frequency response of a parametrically excited system to indicate its stability. Key findings include:
- Unstable regions occur when a secondary frequency response peak merges with the primary resonance peak.
- The primary resonance peak is not at a constant frequency and moves toward lower frequencies as the system approaches an instability.
- Multiple response peaks shift positions as the excitation frequency changes, and their proximity to each other and the primary peak can predict proximity to instability regions.
- Numerical models are used to analyze the transient response of a single-degree-of-freedom parametrically excited system, showing features like shifting resonance peaks that indicate approaching instability.
This Presentation explains about the introduction of Frequency Response Analysis. This video clearly shows advantages and disadvantages of Frequency Response Analysis and also explains frequency domain specifications and derivations of Resonant Peak, Resonant Frequency and Bandwidth.
This document discusses first and second order systems in the s-domain. It defines a first order system as having the highest power of s in the denominator of the transfer function as 1, while a second order system has the highest power of s as 2. Examples of first order systems include velocity of a car and rotating systems. The transfer function of a first order system is τ/(sτ+1). A second order system transfer function is ωn2/(s2+2ξωns+ωn2). Examples of second order systems include mechanical springs and electrical RLC circuits.
This presentation discusses perturbation theory in physical chemistry. Perturbation theory allows solving problems where the potential energy function is slightly different than a problem that can be solved exactly. It provides approximations for how small changes or perturbations to the potential energy function affect the system's energy levels and eigenfunctions. The presentation provides examples of using regular and singular perturbation theory to solve differential equations that model perturbed systems. It also outlines the general process of deriving the first-order energy correction term using perturbation theory.
This document discusses translational and rotational mechanical systems. It begins by defining variables for translational systems like displacement, velocity, acceleration, force, work, and power. It then discusses element laws for translational systems including viscous friction and stiffness elements. The document also introduces rotational systems and defines variables like angular displacement, velocity, acceleration, and torque. It discusses element laws for rotational systems including moment of inertia, viscous friction, and rotational stiffness. Finally, it covers interconnection laws for both translational and rotational systems and provides an example of obtaining the system model for a rotational system.
Engineering Research Publication
Best International Journals, High Impact Journals,
International Journal of Engineering & Technical Research
ISSN : 2321-0869 (O) 2454-4698 (P)
www.erpublication.org
The document discusses first order perturbation theory. It begins by introducing perturbation theory as an approximate method to solve the Schrodinger equation for complex quantum systems where the Hamiltonian cannot be solved exactly. It then presents the key equations of first order perturbation theory. The first order correction to the energy is given by the expectation value of the perturbation operator over the unperturbed ground state wavefunction. The first order correction to the wavefunction is expressed as a linear combination of unperturbed eigenstates.
The stretching modes of several organic molecules were simulated using Gaussian software and compared to observed infrared spectra. Force constants for bonds in acetone, acetonitrile, cyclohexene, methanol, and 1-butanamine were calculated from vibrational frequencies predicted by the quantum mechanical harmonic oscillator model. Plots of observed vs calculated frequencies had slopes near 1, indicating the model accurately predicted frequencies for bonds undergoing stretching vibrations when assuming a pseudo-molecule and direct proportionality between force constant and bond order. However, the model is limited and does not apply to other types of vibrations.
This document discusses analogous systems between mechanical and electrical systems. It introduces force-voltage and force-current analogies where mechanical concepts like force, mass and stiffness are analogous to electrical concepts like voltage, inductance and capacitance. Examples of analogous systems are given for translational, rotational and RLC circuits. The procedure for drawing the mechanical equivalent network of a given mechanical system is also outlined.
This Presentation explains about the introduction of Frequency Response Analysis. This video clearly shows advantages and disadvantages of Frequency Response Analysis and also explains frequency domain specifications and derivations of Resonant Peak, Resonant Frequency and Bandwidth.
This document discusses first and second order systems in the s-domain. It defines a first order system as having the highest power of s in the denominator of the transfer function as 1, while a second order system has the highest power of s as 2. Examples of first order systems include velocity of a car and rotating systems. The transfer function of a first order system is τ/(sτ+1). A second order system transfer function is ωn2/(s2+2ξωns+ωn2). Examples of second order systems include mechanical springs and electrical RLC circuits.
This presentation discusses perturbation theory in physical chemistry. Perturbation theory allows solving problems where the potential energy function is slightly different than a problem that can be solved exactly. It provides approximations for how small changes or perturbations to the potential energy function affect the system's energy levels and eigenfunctions. The presentation provides examples of using regular and singular perturbation theory to solve differential equations that model perturbed systems. It also outlines the general process of deriving the first-order energy correction term using perturbation theory.
This document discusses translational and rotational mechanical systems. It begins by defining variables for translational systems like displacement, velocity, acceleration, force, work, and power. It then discusses element laws for translational systems including viscous friction and stiffness elements. The document also introduces rotational systems and defines variables like angular displacement, velocity, acceleration, and torque. It discusses element laws for rotational systems including moment of inertia, viscous friction, and rotational stiffness. Finally, it covers interconnection laws for both translational and rotational systems and provides an example of obtaining the system model for a rotational system.
Engineering Research Publication
Best International Journals, High Impact Journals,
International Journal of Engineering & Technical Research
ISSN : 2321-0869 (O) 2454-4698 (P)
www.erpublication.org
The document discusses first order perturbation theory. It begins by introducing perturbation theory as an approximate method to solve the Schrodinger equation for complex quantum systems where the Hamiltonian cannot be solved exactly. It then presents the key equations of first order perturbation theory. The first order correction to the energy is given by the expectation value of the perturbation operator over the unperturbed ground state wavefunction. The first order correction to the wavefunction is expressed as a linear combination of unperturbed eigenstates.
The stretching modes of several organic molecules were simulated using Gaussian software and compared to observed infrared spectra. Force constants for bonds in acetone, acetonitrile, cyclohexene, methanol, and 1-butanamine were calculated from vibrational frequencies predicted by the quantum mechanical harmonic oscillator model. Plots of observed vs calculated frequencies had slopes near 1, indicating the model accurately predicted frequencies for bonds undergoing stretching vibrations when assuming a pseudo-molecule and direct proportionality between force constant and bond order. However, the model is limited and does not apply to other types of vibrations.
This document discusses analogous systems between mechanical and electrical systems. It introduces force-voltage and force-current analogies where mechanical concepts like force, mass and stiffness are analogous to electrical concepts like voltage, inductance and capacitance. Examples of analogous systems are given for translational, rotational and RLC circuits. The procedure for drawing the mechanical equivalent network of a given mechanical system is also outlined.
This document discusses steady-state errors in control systems. It defines steady-state error as the difference between the input and output of a system as time approaches infinity. For a unity feedback system, the steady-state error can be calculated from the closed-loop transfer function T(s) or open-loop transfer function G(s). The steady-state error depends on the type of input signal (step, ramp, or parabola) and number of integrations in the system. Systems are classified as Type 0, 1, or 2 based on this number of integrations. The document provides examples of calculating steady-state error for different system types and input signals.
Effects of poles and zeros affect control systemGopinath S
1. A first order system's step response approaches its final value exponentially, determined by the location of its single pole.
2. Adding an additional pole slows the response, as the system is no longer purely first order. However, if the additional pole is far from the original dominant pole, its effect is negligible and the system remains effectively first order.
3. Adding a zero has the opposite effect of a pole - it speeds up the step response. A zero closer to the origin dominates over a pole farther away, making the system response faster than first order.
The document discusses time domain analysis and standard test signals used to analyze dynamic systems. It describes the impulse, step, ramp, and parabolic signals which imitate characteristics of actual inputs such as sudden shock, sudden change, constant velocity, and constant acceleration. The time response of first order systems to these standard inputs is expressed mathematically. The impulse response directly provides the system transfer function. Step response reaches 63% of its final value within one time constant.
This presentation gives complete idea about time domain analysis of first and second order system, type number, time domain specifications, steady state error and error constants and numerical examples.
The document discusses forced vibrations of damped, single degree of freedom linear spring mass systems. It derives the equations of motion for three types of forcing - external forcing, base excitation, and rotor excitation. It presents the steady state solutions and discusses key features, including that the response frequency matches the forcing frequency. The maximum response occurs at resonance when the forcing frequency matches the natural frequency. Engineering applications include designing systems to minimize vibrations by increasing stiffness/natural frequency and damping.
This document discusses dynamic systems and their analysis using transfer functions. It begins by defining dynamic systems as those whose output depends on both current and previous inputs/outputs. It then covers:
- Transfer function representations of linear time-invariant (LTI) systems using Laplace transforms.
- Key properties of transfer functions including poles, zeros and zero-pole-gain form.
- MATLAB representations of transfer functions.
It also defines important concepts for analyzing dynamic systems like time and frequency response, stability, system order, and the effects of pole locations. Specific discussions are included on analyzing first and second order systems.
The document describes a mechanical system project presented by group members Ali Ahssan, Faysal Shahzad, M. Aaqib, and Nafees Ahmed. It discusses translational and rotational mechanical systems. Translational systems move in a straight line and include mass, spring, and dashpot elements. Rotational systems move about a fixed axis and include moment of inertia, dashpot, and torsional spring elements. The document also provides equations to calculate the opposing forces or torques in each element when a force or torque is applied based on Newton's second law of motion.
Introduction to perturbation theory, part-1Kiran Padhy
Perturbation theory provides an approximate method for solving quantum mechanical problems where the Hamiltonian cannot be solved exactly. It involves splitting the Hamiltonian into an exactly solvable unperturbed part (H0) and a perturbed part (H1) treated as a small disturbance. The eigenvalues and eigenstates of the full Hamiltonian are expressed as power series expansions in terms of the perturbation strength parameter λ, allowing the effects of the perturbation to be calculated order by order. There are two types of perturbation theory: time-independent, where the unperturbed eigenstates are stationary; and time-dependent, where they vary with time under the perturbation.
The document discusses different types of inputs to control systems including impulse, step, ramp, and parabolic inputs. It analyzes the time response and steady state error of systems subjected to these different inputs. For first order systems, it derives the transfer function for a simple RC circuit and describes the transient response. For second order systems, it defines natural frequency, damping ratio, and damping cases. It also lists specifications for transient response including delay time, rise time, peak time, setting time, and peak overshoot.
1. The document discusses time domain analysis of second order systems. It defines key terms like damping ratio, natural frequency, and describes the four categories of responses based on damping ratio: overdamped, underdamped, undamped, and critically damped.
2. An example shows how to determine the natural frequency and damping ratio from a given transfer function. The poles of a second order system depend on these parameters.
3. The time domain specification of a second order system's step response is explained, including definitions of delay time, rise time, peak time, settling time, and overshoot.
This document discusses the quantum theory of light dispersion using time-dependent perturbation theory. It describes how bound electrons in materials contribute to the permittivity and optical properties when subjected to an external electric field. The perturbation leads to polarization of electron orbitals and possible transitions to excited states. Absorption of light occurs when the photon energy matches the energy difference between bound states. The permittivity and refractive index are derived in terms of oscillator strengths, and dispersion is explained through resonant absorption at certain photon frequencies.
This document discusses mechanical vibrations, which occur when a system oscillates around an equilibrium position. It provides definitions for key vibration terms like period, frequency, amplitude, damping, and forced versus free vibrations. As examples, it examines the simple harmonic motion of a mass attached to a spring, and the motion of a simple pendulum through both approximate and exact solutions.
The document summarizes research on modeling the resonant frequency of MEMS relays and switches. It describes a simple mass-spring model used to simulate the resonant switch and analyze the cantilever system's behavior at different driving frequencies. When the frequency matches resonance, the displacement amplitude grows continuously until contact is hit, at which point the oscillation becomes nonlinear. Next steps include revising the model to account for energy loss and other vibration modes, preventing phase shifts on contact, converting it to a physical contact model, and testing prototype devices.
The document defines transfer function as the ratio of the Laplace transform of the output to the input of a system with zero initial conditions. It discusses poles and zeros, which are values of s that make the transfer function tend to infinity or zero. Strictly proper, proper, and improper transfer functions are classified based on the order of the numerator and denominator polynomials. The characteristic equation is obtained by equating the denominator of the transfer function to zero. Advantages of transfer functions include representing systems with algebraic equations and determining poles, zeros and differential equations. Translational and rotational mechanical systems are described along with their resisting forces, and D'Alembert's principle is explained.
This document contains notes from lectures 23-24 on time response and steady state errors in discrete time control systems. It begins with an outline of the lecture topics, then provides introductions and examples related to time response, the final value theorem, and steady state errors. It defines concepts like position and velocity error constants and shows examples of calculating steady state error for different system transfer functions. The document contains MATLAB examples and homework problems related to analyzing discrete time systems.
Here are the steps to solve for the transfer function G(s) = X2(s)/F(s) for the given system:
1. Draw the free body diagrams for both masses M1 and M2 showing all the forces acting on each mass.
2. Write the Newton's second law equation for M1:
(M1s2 + f1vs + k1)X1(s) - k2(X1(s) - X2(s)) = 0
3. Write the Newton's second law equation for M2:
-k2(X1(s) - X2(s)) + (M2s2 + f2vs + k
This document summarizes classical dynamics and small amplitude oscillations. It discusses oscillatory motion near equilibrium positions and developing the theory using Lagrange's equations. Normal modes of coupled oscillating systems are explored, where the normal coordinates represent eigenvectors that oscillate at characteristic frequencies. The principles of superposition and matrix representations are used to analyze examples like two coupled pendulums and a system of two masses connected by three springs.
This document summarizes key concepts in vibration of single-degree-of-freedom (SDOF) systems. It discusses the generalized model of SDOF systems and provides examples. It then covers the differential equations of motion for SDOF systems using Newton's law and the energy method in the time domain. Specific examples are given for mass-spring, simple pendulum, and cantilever beam systems. Considerations for equivalent mass and stiffness of springs are also addressed.
Mr. C.S.Satheesh, M.E.,
Frequency response analysis
Frequency Domain Specifications
Resonant Peak Mr
Resonant Frequency ωr
Bandwidth ωh
Cut – off Rate
Gain margin Kg
Phase margin γ
POLAR PLOT
Bode PLOT
The document presents an analytical model of a single-degree-of-freedom mechanical system subject to parametric excitation through a periodically varying stiffness. Numerical simulations show unbounded vibration occurring at the natural frequency and twice the natural frequency. Increasing the stiffness fluctuation amplitude produces additional unbounded response regions. The output frequency is observed to fluctuate as the input frequency sweeps through resonance regions, unlike a system with constant stiffness. The model provides insight into parametric instability in gear systems, where tooth contact stiffness varies periodically during operation.
Este documento describe los sistemas operativos. Un sistema operativo es el software básico que proporciona una interfaz entre los programas, el hardware y el usuario, administrando los recursos de la máquina y organizando archivos. Los sistemas operativos más comunes son Windows, Mac, Linux y Android. Existen dos tipos principales: los sistemas operativos propietarios como Windows, que tienen limitaciones de uso y modificación, y los sistemas operativos gratuitos como Linux, que permiten la libertad de usar, estudiar, distribuir y mejorar el software.
This document discusses steady-state errors in control systems. It defines steady-state error as the difference between the input and output of a system as time approaches infinity. For a unity feedback system, the steady-state error can be calculated from the closed-loop transfer function T(s) or open-loop transfer function G(s). The steady-state error depends on the type of input signal (step, ramp, or parabola) and number of integrations in the system. Systems are classified as Type 0, 1, or 2 based on this number of integrations. The document provides examples of calculating steady-state error for different system types and input signals.
Effects of poles and zeros affect control systemGopinath S
1. A first order system's step response approaches its final value exponentially, determined by the location of its single pole.
2. Adding an additional pole slows the response, as the system is no longer purely first order. However, if the additional pole is far from the original dominant pole, its effect is negligible and the system remains effectively first order.
3. Adding a zero has the opposite effect of a pole - it speeds up the step response. A zero closer to the origin dominates over a pole farther away, making the system response faster than first order.
The document discusses time domain analysis and standard test signals used to analyze dynamic systems. It describes the impulse, step, ramp, and parabolic signals which imitate characteristics of actual inputs such as sudden shock, sudden change, constant velocity, and constant acceleration. The time response of first order systems to these standard inputs is expressed mathematically. The impulse response directly provides the system transfer function. Step response reaches 63% of its final value within one time constant.
This presentation gives complete idea about time domain analysis of first and second order system, type number, time domain specifications, steady state error and error constants and numerical examples.
The document discusses forced vibrations of damped, single degree of freedom linear spring mass systems. It derives the equations of motion for three types of forcing - external forcing, base excitation, and rotor excitation. It presents the steady state solutions and discusses key features, including that the response frequency matches the forcing frequency. The maximum response occurs at resonance when the forcing frequency matches the natural frequency. Engineering applications include designing systems to minimize vibrations by increasing stiffness/natural frequency and damping.
This document discusses dynamic systems and their analysis using transfer functions. It begins by defining dynamic systems as those whose output depends on both current and previous inputs/outputs. It then covers:
- Transfer function representations of linear time-invariant (LTI) systems using Laplace transforms.
- Key properties of transfer functions including poles, zeros and zero-pole-gain form.
- MATLAB representations of transfer functions.
It also defines important concepts for analyzing dynamic systems like time and frequency response, stability, system order, and the effects of pole locations. Specific discussions are included on analyzing first and second order systems.
The document describes a mechanical system project presented by group members Ali Ahssan, Faysal Shahzad, M. Aaqib, and Nafees Ahmed. It discusses translational and rotational mechanical systems. Translational systems move in a straight line and include mass, spring, and dashpot elements. Rotational systems move about a fixed axis and include moment of inertia, dashpot, and torsional spring elements. The document also provides equations to calculate the opposing forces or torques in each element when a force or torque is applied based on Newton's second law of motion.
Introduction to perturbation theory, part-1Kiran Padhy
Perturbation theory provides an approximate method for solving quantum mechanical problems where the Hamiltonian cannot be solved exactly. It involves splitting the Hamiltonian into an exactly solvable unperturbed part (H0) and a perturbed part (H1) treated as a small disturbance. The eigenvalues and eigenstates of the full Hamiltonian are expressed as power series expansions in terms of the perturbation strength parameter λ, allowing the effects of the perturbation to be calculated order by order. There are two types of perturbation theory: time-independent, where the unperturbed eigenstates are stationary; and time-dependent, where they vary with time under the perturbation.
The document discusses different types of inputs to control systems including impulse, step, ramp, and parabolic inputs. It analyzes the time response and steady state error of systems subjected to these different inputs. For first order systems, it derives the transfer function for a simple RC circuit and describes the transient response. For second order systems, it defines natural frequency, damping ratio, and damping cases. It also lists specifications for transient response including delay time, rise time, peak time, setting time, and peak overshoot.
1. The document discusses time domain analysis of second order systems. It defines key terms like damping ratio, natural frequency, and describes the four categories of responses based on damping ratio: overdamped, underdamped, undamped, and critically damped.
2. An example shows how to determine the natural frequency and damping ratio from a given transfer function. The poles of a second order system depend on these parameters.
3. The time domain specification of a second order system's step response is explained, including definitions of delay time, rise time, peak time, settling time, and overshoot.
This document discusses the quantum theory of light dispersion using time-dependent perturbation theory. It describes how bound electrons in materials contribute to the permittivity and optical properties when subjected to an external electric field. The perturbation leads to polarization of electron orbitals and possible transitions to excited states. Absorption of light occurs when the photon energy matches the energy difference between bound states. The permittivity and refractive index are derived in terms of oscillator strengths, and dispersion is explained through resonant absorption at certain photon frequencies.
This document discusses mechanical vibrations, which occur when a system oscillates around an equilibrium position. It provides definitions for key vibration terms like period, frequency, amplitude, damping, and forced versus free vibrations. As examples, it examines the simple harmonic motion of a mass attached to a spring, and the motion of a simple pendulum through both approximate and exact solutions.
The document summarizes research on modeling the resonant frequency of MEMS relays and switches. It describes a simple mass-spring model used to simulate the resonant switch and analyze the cantilever system's behavior at different driving frequencies. When the frequency matches resonance, the displacement amplitude grows continuously until contact is hit, at which point the oscillation becomes nonlinear. Next steps include revising the model to account for energy loss and other vibration modes, preventing phase shifts on contact, converting it to a physical contact model, and testing prototype devices.
The document defines transfer function as the ratio of the Laplace transform of the output to the input of a system with zero initial conditions. It discusses poles and zeros, which are values of s that make the transfer function tend to infinity or zero. Strictly proper, proper, and improper transfer functions are classified based on the order of the numerator and denominator polynomials. The characteristic equation is obtained by equating the denominator of the transfer function to zero. Advantages of transfer functions include representing systems with algebraic equations and determining poles, zeros and differential equations. Translational and rotational mechanical systems are described along with their resisting forces, and D'Alembert's principle is explained.
This document contains notes from lectures 23-24 on time response and steady state errors in discrete time control systems. It begins with an outline of the lecture topics, then provides introductions and examples related to time response, the final value theorem, and steady state errors. It defines concepts like position and velocity error constants and shows examples of calculating steady state error for different system transfer functions. The document contains MATLAB examples and homework problems related to analyzing discrete time systems.
Here are the steps to solve for the transfer function G(s) = X2(s)/F(s) for the given system:
1. Draw the free body diagrams for both masses M1 and M2 showing all the forces acting on each mass.
2. Write the Newton's second law equation for M1:
(M1s2 + f1vs + k1)X1(s) - k2(X1(s) - X2(s)) = 0
3. Write the Newton's second law equation for M2:
-k2(X1(s) - X2(s)) + (M2s2 + f2vs + k
This document summarizes classical dynamics and small amplitude oscillations. It discusses oscillatory motion near equilibrium positions and developing the theory using Lagrange's equations. Normal modes of coupled oscillating systems are explored, where the normal coordinates represent eigenvectors that oscillate at characteristic frequencies. The principles of superposition and matrix representations are used to analyze examples like two coupled pendulums and a system of two masses connected by three springs.
This document summarizes key concepts in vibration of single-degree-of-freedom (SDOF) systems. It discusses the generalized model of SDOF systems and provides examples. It then covers the differential equations of motion for SDOF systems using Newton's law and the energy method in the time domain. Specific examples are given for mass-spring, simple pendulum, and cantilever beam systems. Considerations for equivalent mass and stiffness of springs are also addressed.
Mr. C.S.Satheesh, M.E.,
Frequency response analysis
Frequency Domain Specifications
Resonant Peak Mr
Resonant Frequency ωr
Bandwidth ωh
Cut – off Rate
Gain margin Kg
Phase margin γ
POLAR PLOT
Bode PLOT
The document presents an analytical model of a single-degree-of-freedom mechanical system subject to parametric excitation through a periodically varying stiffness. Numerical simulations show unbounded vibration occurring at the natural frequency and twice the natural frequency. Increasing the stiffness fluctuation amplitude produces additional unbounded response regions. The output frequency is observed to fluctuate as the input frequency sweeps through resonance regions, unlike a system with constant stiffness. The model provides insight into parametric instability in gear systems, where tooth contact stiffness varies periodically during operation.
Este documento describe los sistemas operativos. Un sistema operativo es el software básico que proporciona una interfaz entre los programas, el hardware y el usuario, administrando los recursos de la máquina y organizando archivos. Los sistemas operativos más comunes son Windows, Mac, Linux y Android. Existen dos tipos principales: los sistemas operativos propietarios como Windows, que tienen limitaciones de uso y modificación, y los sistemas operativos gratuitos como Linux, que permiten la libertad de usar, estudiar, distribuir y mejorar el software.
Cúcuta es la ciudad natal de la autora, y cuenta con varios sitios turísticos importantes como la Casa de la Bagatela, donde funcionó el poder ejecutivo de la Gran Colombia en 1821, la Casa de Santander donde vivió los primeros 13 años el político Francisco de Paula Santander, y la Catedral de San José de Cúcuta construida entre 1889 y 1956. Otros lugares notables son el Malecón con restaurantes y zonas deportivas, el Templo Histórico de la Gran Colombia con estilo neoclásico, los centros comerciales
El documento habla sobre la práctica de tres valores: solidaridad, respeto y puntualidad. Define cada valor de la siguiente manera: la solidaridad es apoyar causas ajenas en situaciones difíciles, el respeto es tratar a los demás y sus opiniones con consideración, y la puntualidad es coordinarse cronológicamente para cumplir tareas a tiempo. También incluye enlaces a videos relacionados con estos temas.
El documento describe tres valores: solidaridad, respeto y puntualidad. La solidaridad implica compartir la situación de otros y brindar apoyo desinteresado. El respeto significa no menospreciar a los demás, ser considerado con sus sentimientos y tratarlos con dignidad. La puntualidad se refiere a coordinarse para cumplir tareas y obligaciones en el tiempo acordado.
Timothy Sisk has over 15 years of experience in software testing, management, and implementation. He currently works as a Quality Assurance Lead, coordinating testing efforts for internally developed and third party applications, including testing for ICD10 and a third party EHR system. Previously he held roles as a Senior Quality Assurance Analyst and Software Quality Assurance Analyst, developing and executing test plans and procedures.
Eight lessons are provided for improving online fundraising in 2015 based on analysis of 2014 results. Key insights include: 1) Personalizing outreach based on donor relationships and data; 2) Leveraging high-performing campaigns through resending and follow-ups; 3) Creating fundraising campaigns around additional holidays and events; 4) Framing donations as purchases by highlighting tangible impacts; 5) Motivating teams and donors through competitions; 6) Optimizing the donor experience for mobile; 7) Providing inspiring non-ask content to engage donors; 8) Thanking and recognizing donors for their impact. Overall, testing and analyzing donor behaviors and preferences is emphasized to best respond to their needs.
Este documento presenta dos problemas de tablas numéricas para resolver en clase. El primer problema involucra a tres niñas (Paola, Sofía y Diana) que tienen 30 prendas de vestir en total, de las cuales 15 son blusas. Se proporciona información adicional sobre la cantidad de diferentes tipos de prendas que tiene cada niña. El segundo problema involucra a tres casas (de Talía, Paulina y Belén) que tienen un total de 16 animales domésticos entre perros, gatos, canarios y loros. Se proporciona información sobre la
Haiku Deck is a presentation tool that allows users to create Haiku-style slideshows. The tool encourages users to get started making their own Haiku Deck presentations, which can be shared on SlideShare. In just 3 sentences, it promotes creating Haiku Deck presentations and publishing them to SlideShare.
Converse seeks to increase sales of its top-selling Chuck Taylor All-Star sneakers, which bring in $1.3 billion annually, by launching a YouTube video series featuring famous musicians like Rihanna, Ariana Grande, and Maroon 5 to appeal to their target demographic of men and women aged 16-24. The proposed $3.5 million marketing budget would pay the musicians between $250,000-$750,000 each for their participation and dedicate over $1 million to video creation and maintaining the blog, AdWords, and social media promotion for the series.
Campbell's Soup aimed to create a digital campaign targeting young adults to increase brand loyalty and awareness. The $10 million budget was allocated to social media, Google Adwords, and inbound marketing. The campaign sought to engage college students and young professionals through a "Cool Mom" approach on social media, cooking videos, recipes, mobile apps, and contests in order to portray Campbell's Soup as suitable for the younger generation.
This short document promotes creating presentations using Haiku Deck, a tool for making slideshows. It encourages the reader to get started making their own Haiku Deck presentation and sharing it on SlideShare. In just one sentence, it pitches the idea of using Haiku Deck to easily create engaging slideshow presentations.
The document discusses the formation of cooperative credit societies to address the financial needs of employees. It notes that employees often face economic hardship due to low wages that do not cover basic necessities. Cooperative credit societies can help by providing cheap credit and supplying goods at fair prices. The key principles of cooperatives are voluntary membership, democratic control, and distributing surplus funds to members. Cooperatives are financed through non-repayable fees, transferable shares, and deposits from members. Forming credit cooperatives allows employees to assist each other by addressing their shared need for affordable credit.
Las tutorías tienen como objetivo reforzar los conocimientos de los alumnos a través de casos prácticos seleccionados aleatoriamente por el coordinador del curso. Los tutores deben acordar las fechas de las sesiones con el coordinador, asistir a las reuniones para recibir las pautas de los casos, comunicar los horarios disponibles a los alumnos, llevar control de asistencia, y preparar informes de horas realizadas para obtener créditos extraacadémicos. No está permitido entregar soluciones a los al
Time Response Analysis of system
Standard Test Signals
What is time response ?
Types of Responses
Analysis of First order system
Analysis of Second order system
This document contains a question bank for control systems with questions about various control system concepts. Some key points:
1. It defines open loop and closed loop control systems, with closed loop systems having feedback to correct errors and maintain desired output values.
2. The components of a feedback control system are identified as the plant, feedback path, error detector, actuator and controller.
3. Different types of controllers are listed, including proportional, PI, PD and PID controllers. The proportional controller produces an output proportional to the error signal.
Transient and Steady State Response - Control Systems EngineeringSiyum Tsega Balcha
. Two crucial aspects of this behavior are transient and steady-state responses. These concepts encapsulate how a system behaves over time, from the moment an input is applied to when the system settles into a stable state. The transient response of a system characterizes its behavior during the initial phase after a change in input. It reflects how the system reacts as it transitions from one state to another. This phase is marked by dynamic changes in the system's output as it adjusts to the new conditions imposed by the input.
Characteristics of Transient Response are Time Constant, overshoot, settling time and damping.
Once the transient effects have subsided, the system enters the steady-state, where its behavior becomes constant over time. In this phase, the system operates under stable conditions, and its output remains within a narrow range around the desired value, despite fluctuations in input or external disturbances. Characteristics of Steady-State Response are Steady-State Error, stability, accuracy, robustness,.
This project developed a control system to continuously measure the quality factor (Q) of mechanical oscillators. The system locks the phase between the oscillator's exciter and normal mode to π/2 and locks the oscillator's amplitude with control loops. This allows the rate of energy input to equal the rate of energy loss, from which Q can be determined. The author has successfully locked both phase and amplitude to within fractions of a percent on a test oscillator with Q of 5×104 ± 104 using PID controllers. The system shows promise for efficiently measuring Q of LIGO test masses with Qs up to 8×105 ±105. Current work is adapting the PID controllers for this higher-Q system.
Mr. C.S.Satheesh, M.E.,
Time Response in systems
Time Response
Transient response
Steady-state response.
Delay Time (td)
Rise Time (tr)
Peak Time (tp)
Maximum Overshoot (Mp)
Settling Time (tS)
Standard Test Signals
Impulse signal
Step signal
Ramp signal
Parabolic signal
RF and Optical Realizations of Chaotic DynamicsEmeka Ikpeazu
This document discusses two systems that exhibit chaos: the Josephson junction and optical systems. It describes how the Josephson junction, a quantum device made of two superconductors separated by an insulator, can display chaotic behavior when coupled to an RLC circuit. It also explains how feedback in laser systems can induce optical chaos characterized by spectral broadening. The document emphasizes that while chaos is interesting, controlling and harnessing chaos in these systems could enable applications in areas like signal processing and communications.
The document discusses nonlinear systems and their analysis. Some key points:
1) All physical systems are inherently nonlinear, and nonlinearity introduces difficulties in analysis compared to linear systems. Nonlinear systems do not obey superposition and can exhibit phenomena like limit cycles and chaos.
2) Mathematical models of nonlinear systems involve state equations where the states are functions of prior states, inputs, and time. Phase plane analysis is a graphical technique used to study nonlinear dynamical systems.
3) Analyzing nonlinear systems is challenging as techniques like linearization only approximate local behavior. Phenomena like limit cycles, subharmonics, and multiple equilibria depend on initial conditions in complex ways. Stability also differs from linear systems
REPORT SUMMARYVibration refers to a mechanical.docxdebishakespeare
REPORT SUMMARY
Vibration refers to a mechanical phenomenon involving oscillations about a point. These oscillations can be of any imaginable range of amplitudes and frequencies, with each combination having its own effect. These effects can be positive and purposefully induced, but they can also be unintentional and catastrophic. It's therefore imperative to understand how to classify and model vibration.
Within the classroom portion of ME 345, we discussed damped and undamped vibrations, appropriate models, and several of their properties. The purpose of Lab 3 is to give us the corresponding "hands-on" experience to cement our understanding of the theory.
As it turns out, vibration can be modeled with a simple spring-mass system (spring-mass-damper system for damped vibration). In order to create a mathematical model for our simple spring-mass system, we apply Newton's second law and sum the forces about the mass. After applying some of our knowledge of differential equations, the result is a second order linear differential equation (in vector form). This can easily be converted to the scalar version, from which it's easy to glean various properties of the vibration (i.e. natural frequency, period, etc.).
In the lab, we were provided with a PASCO motion sensor, USB link, ramp, and accompanying software. All of the aforementioned equipment was already assembled and connected. The ramp was set up at an angle with a stop on the elevated end and the motion sensor on the lower end. The sensor was connected to the USB link, which was in turn connected to the computer. We chose to use the Xplorer GLX software to interface with the sensor and record our data. After receiving our equipment, we gathered data on our spring's extension with a known load to derive a spring constant. We were provided with a small cart to which we attached weights to increase its mass. In order to model free vibration, we placed the cart on the track and attached it to the stop at the top of the ramp with a spring. After displacing the cart a certain distance from its equilibrium point, the cart was released and was allowed to oscillate on the track while we recorded its distance from the sensor. This was done with displacements of -20cm, -10cm, +10cm, and +20cm from the system's equilibrium point. After gathering this data for the "free" case, a magnet was attached to the front of the car, spaced as far from the track as possible. As the track is magnetic, this caused a slight damping effect, basically converting our spring-mass system to an underdamped spring-mass-damper system. After repeating the procedure for the "free" case, we moved the magnets as close to the track as possible (causing the system to become overdamped) and again repeated the procedure for the "free" case.
We were finally able to determine the period, phase angle, damping coefficients, and circular and cyclical frequencies for the three systems. There were similarities and differ ...
This document discusses time response analysis of control systems. It begins by introducing poles and zeros and their influence on system response. It then examines the transient response of first-order and second-order systems. Key concepts covered include time constant, rise time, settling time, damping ratio, natural frequency, and the different responses based on being underdamped, overdamped or critically damped. The document provides examples of step responses for different system types and calculations of critical response specifications.
1) The document discusses dynamic characteristics of measurement systems, focusing on zero-order, first-order, and second-order systems.
2) First-order systems exhibit an exponential response to a step input, reaching 63.2% of the final value after one time constant. The time constant can be determined experimentally from the step response.
3) Second-order systems can exhibit underdamped, critically damped, or overdamped responses depending on the damping ratio. The natural frequency and damping ratio characterize the system's dynamic behavior.
ME-314 Introduction to Control Engineering is a course taught to Mechanical Engineering senior undergrads. The course is taught by Dr. Bilal Siddiqui at DHA Suffa University. This lecture is about time response of systems derived by inspection of poles and zeros. First and second order systems are considered, along with higher order and nonminimum phase systems
The document provides an overview of seismic sensors and their calibration. It discusses two main types of seismic sensors - inertial seismometers which measure ground motion relative to a suspended mass, and strainmeters which measure the relative motion of two points in the ground. Inertial seismometers are generally more sensitive but strainmeters can outperform them at very low frequencies. The document focuses on describing the dynamic properties and calibration of inertial seismometers. There are four equivalent ways to characterize the response of a linear seismometer system: differential equations, transfer functions, complex frequency response, and impulse response. Calibration involves determining the system's response and expressing it using one of these representations.
ER Publication,
IJETR, IJMCTR,
Journals,
International Journals,
High Impact Journals,
Monthly Journal,
Good quality Journals,
Research,
Research Papers,
Research Article,
Free Journals, Open access Journals,
erpublication.org,
Engineering Journal,
Science Journals,
Small-Signal (or Small Disturbance) Stability is the ability of a power system to maintain synchronism when subjected to small disturbances
such disturbances occur continually on the system due to small variations in loads and generation
disturbance considered sufficiently small if linearization of system equations is permissible for analysis
Corresponds to Liapunov's first method of stability analysis
Small-signal analysis using powerful linear analysis techniques provides valuable information about the inherent dynamic characteristics of the power system and assists in its robust design
1) The document studies the noise sensitivity of balancing tasks modeled as an inverted pendulum with delayed feedback control.
2) It considers two control strategies - act and wait control with intermittent feedback, and continuous feedback control governed by a switching manifold.
3) For act and wait control, stability is maximized near "deadbeat" control parameters, but noise can still cause fluctuations during waiting periods. Continuous feedback control exhibits bistability near bifurcation points, making it sensitive to noise near these points.
ELEG 421 Control Systems Transient and Steady State .docxtoltonkendal
ELEG 421
Control Systems
Transient and Steady State
Response Analyses
Dr. Ashraf A. Zaher
American University of Kuwait
College of Arts and Science
Department of Electrical and Computer Engineering
Layout
2
Objectives
This chapter introduces the analysis of the time response of different
control systems under different scenarios. Only first and second order
systems will be considered in details using analytical and numerical
methods. Extension to higher order systems will be developed. Both
transient and steady state responses will be evaluated. Stability analysis
will be analyzed for different kinds of feedback, while investigating the
effect of both proportional and derivative control actions on the
performance of the closed-loop system. Finally systems types and
steady state errors will be calculated for unity feedback.
Outcomes
By the end of this chapter, students will be able to:
evaluate both transient/steady state responses for control systems,
analyze the stability of closed-loop LTI systems,
investigate the effect of P and I control actions on performance, and
understand dominant dynamics of higher order systems.
Dr. Ashraf Zaher
Introduction
3
Test signals
Transient response
Steady state response
Analytical techniques, and
Numerical (simulation) techniques.
Stability (definition and analysis methods),
Relative stability, and
Effect of P/I control actions on stability and performance.
Summary of the used systems:
First order systems,
Second order systems, and
Higher order systems.
Dr. Ashraf Zaher
Test Signals
4 Dr. Ashraf Zaher
Impulse function:
Used to simulate shock inputs,
Laplace transform: 1.
Step function:
Used to simulate sudden disturbances,
Laplace transform: 1/s.
Ramp function:
Used to simulate gradually changing inputs,
Laplace transform: 1/s2.
Sinusoidal function(s):
Used to test response to a certain frequency,
Laplace transform: s/(s2+ω2) for cos(ωt) and ω/(s2+ω2) for sin(ωt).
White noise function:
Used to simulate random noise,
It is a stochastic signal that is easier to deal with in the time domain.
Total response:
C(s) = R(s)*TF(s) = Ctr(s) + Css(s) → c(t) = ctr(t) + css(t)
Fundamentals
5 Dr. Ashraf Zaher
Definitions:
Zeros (Z) of the TF
Poles (P) of the TF
Transient Response (Natural)
Steady State Response (Forced)
Total Response
Limits:
Initial values
Final values
Systems (?Zs):
First order (one P)
Second order (two Ps)
Higher order!
More:
Stability and relative stability
Steady state errors (unity feedback)
First Order Systems
6 Dr. Ashraf Zaher
TF:
T: time constant
Unit Step Response:
1
1
)(
)(
+
=
TssR
sC
)/1(
11
1
1
1
11
)(
TssTs
T
sTss
sC
+
−=
+
−=
+
=
Ttetc /1)( −−=
632.01)( 1 =−== −eTtc
T
e
Tdt
tdc Tt
t
11)( /
0
== −
=
01)0( 0 =−== etc
11)( =−=∞= −∞etc
First Order Systems.
Time response of discrete systems 4th lecturekhalaf Gaeid
1. The document discusses the time response of discrete-time systems, including their transient and steady-state response. It describes parameters for characterizing transient response such as rise time, delay time, peak time, and settling time.
2. Steady-state errors are also examined for different system types (Type-0, 1, 2 systems) and inputs (step, ramp, parabolic). Examples are provided to calculate steady-state errors.
3. The response of discrete-time systems is derived using impulse response sequences and convolution sums. The time response is broken into zero-input and zero-state responses.
Oscillatory Behaviors of Second Order Forced Functional Differential Equation.IOSR Journals
Oscillatory behaviors of second order forced functional differential equation is considered. The
oscillation of this equation is shown to be maintained under the effect of certain forcing terms, and the
oscillatory equation can serve as mathematical tool for simulation of processes and phenomina observed in
control theory.
Oscillatory Behaviors of Second Order Forced Functional Differential Equation.
ASME_2015_VIB_Paper_Draft
1. Proceedings of the ASME 2015 International Design Engineering Technical Conferences &
Computers and Information in Engineering Conference
IDETC/CIE 2015
August 2-5, 2015, Boston, MA, United States
DETC2015-47715
DRAFT: TRANSIENT FREQUENCY RESPONSE BEHAVIOR AS AN INDICATOR OF
STABILITY IN PARAMETRICALLY EXCITED SYSTEMS
Johnathan Losek
Mechanical Engineering Student
York College of Pennsylvania
York, PA 17403
jlosek@ycp.edu
Tristan M. Ericson∗
Assistant Professor
Department of Physical Sciences
York College of Pennsylvania
York, PA 17403
tericson@ycp.edu
ABSTRACT
Parametrically excited systems are often analyzed for sta-
bility without considering the response in the time and frequency
domains. This work, however, looks more closely at the tran-
sient response, particularly in the frequency domain, as the re-
sponse of a parametrically excited time-variant system settles to
equilibrium (stable) or approaches an unbounded response (un-
stable). There are a number of peaks in the frequency domain,
other than the primary resonance near the natural frequency,
that shift along the frequency axis through a sweep. A number
of unique features of this behavior are considered in light of the
stability properties of the system. Unstable regions occur when
one of the secondary peaks merges with the primary resonance.
The primary resonance is not at a constant frequency; it moves
asymptotically near instability regions. The research is driven by
numerical analyses, but experimental and analytical validation
techniques will be discussed in the presentation. This research
shows that multiple response peaks are important in paramet-
ric response. Their proximity to the primary resonance indicates
how close the operating frequency of the system is to an insta-
bility region. This form of analysis can be used in practical ap-
plications to predict how close a stable system is to an unstable
response if a transient signal can be measured.
∗Address all correspondence to this author.
INTRODUCTION
Mechanical system vibration depends on the parameters of
the system and the source of excitation. Parametric excitation is
a type of stimulus that commonly occurs in gear systems [1–3].
Gears are important components in many commercial and mili-
tary applications, but they are vulnerable to vibration issues be-
cause tooth contact fluctuates during operation. This periodic
fluctuation results in a time varying tooth stiffness. Large am-
plitude vibration occurs at particular parametrically excited fre-
quencies leading to instability. Current analytical models provide
solutions for frequencies at which parametric instability occurs in
geared systems [4] and general dyanmic systems [5,6], but work
considering transient response behavior deeply is limited.
An analytical model is developed for a single-degree-of-
freedom system with sinusoidal time varying stiffness. Numeri-
cal result are obtained in the frequency and time domains. Fre-
quency response plots show unbounded response behavior oc-
curring at twice the natural frequency, the natural frequency, and
lower frequencies as predicted by current models. Unbounded
regions away from the natural and twice the natural frequency
are detected as the amplitude of stiffness fluctuation is increased.
The domain of an unbounded response region increases linearly
with increasing amplitude of stiffness fluctuation. Analysis of the
frequency spectrum reveals multiple response peaks. Unbounded
response occurs as sub-peaks coalesce with the primary output
frequency.
1 Copyright c 2015 by ASME
2. ANALYTICAL MODEL
An analytical model of a single-degree-of-freedom, mass-
spring-damper system with time varying stiffness
k(t) = k +αksin(ωwt) (1)
is used in this analysis where ωe is the parametric excitation fre-
quency. The amplitude of the fluctuating portion of the stiffness
is defined as a percentage of the mean value k by the nondomen-
sional parameter 0 ≤ α ≤ 0. The equation of motion for the para-
metrically excited system with no external excitation is
m
d2x
dt2
+c
dx
dt
+kx+αksin(ωet)x = 0. (2)
Dimensionless parameters are obtained for convenience. Let
x(t) = ε (τ(t))xc and τ(t) = t
tc
, where ε is the nondimensional
displacement, τc is the nondimensional time, and xc and Tc are
constants. Thus, the derivatives of x are
dx
dt
=
dε
dτ
dτ
dt
xc =
dε
dτ
xc
tc
(3)
and
d2x
dt
=
d2ε
dτ2
d2τ
dt2
xc =
d2ε
dτ2
xc
tc
. (4)
Substituting these dimensionless variables into the equation
of motion, we have
d2ε
dτ2
+2ζωntc
dε
dτ
+ω2
nt2
c ε +αω2
nt2
c sin(ωτtc) = 0 (5)
where ζ is the damping ratio and ωn is the natural frequency.
Letting ω2
nt2
c = 1, or tc = 1
ωn
, the nondimansional equation
¨ε(τ)+2ζ ˙ε(τ)+ε(τ) = −α sin
ω
ωn
τ ε(τ) (6)
is used in the numerical analysis.
NUMERICAL RESULTS
A numerical solver is used to obtain the transient response
of the parametrically excited time-variant system represented in
equation 6 due to an initial displacement of εo = 1. The solver
gives nondimensionalized displacement ε in the time domian.
The simulation is run through a frequency sweep from ωe
ωn
= 0.5
to ωe
ωn
= 3.0. An FFT gives the frequency spectrum for each so-
lution set. Even though the transient signals die out (or grow
unbounded within instability regions), their frequency content
shows some interesting behavior. First, there is typically a pri-
mary response peak near the natural frequency, but it is not at the
natural frequency, and it is not constant. Figure 1 shows the spec-
trum zoomed in near the primary response peak for two different
parametric excitation frequencies. The primary peak can appear
below or above the system natural frequency. In subsequent anal-
ysis, we see that the primary response peak dips to lower fre-
quencies before instability regions (during a speed sweep) and
appears at higher frequencies after the sweep emerges from the
instability region.
0
0.01
0.02
0.03
0.04
0.05
0.6 0.8 1 1.2 1.4
ε
ω
nω
ω
nω =1.03eω
nω =0.87e
FIGURE 1: Frequency spectrum of the stable transient response
zoomed in near the primary response peak with ζ = 0.01, α =
0.80 and (solid black) ωe
ωn
= 0.87 and (dashed red) ωe
ωn
= 1.03
Continuing with frequency domain analyis of the transient
response, there are a number of peaks other than the primary re-
sponse peak. As system damping ζ is decreased, the number of
observable peaks increases. With zero damping, there may be
infinite number of them. Figure 2 shows the transient response
when the excitation frequency ωe = 1.60ωn. At this point, the
frequency sweep is past the instability region around ωn and it
is approaching the largest instability region near 2ωn typical of
parametric systems. The figure shows that there are several fre-
quency peaks besides the primary peak ωp, which is found near
the natural frequency ωn. The arrows indicate the direction that
2 Copyright c 2015 by ASME
3. these peaks are moving with the increasing speed sweep. The
system is currently stable, but it is approacing the instability re-
gion around 2ωn. As the parametric excitation frequency is in-
creased, the peaks will move in the directions indicated. The
instability region is entered with three simultaneous events:
1. The lowest peak (below ω
ωn
= 0.5) reaches zero.
2. The peak just below ωp collides with ωp.
3. The peak just below ω
ωn
= 1.5 catches up with the next peak
above ω
ωn
= 1.5 (because the former is moving faster than
the latter).
The combined peaks move together, to higher frequencies,
throughout the instability regions. The system becomes stable
again when the peaks separate. Thus, the proximity of peaks to
one another, and particularly a secondary peak to the primary
peak, indicats closeness to an instability region. This is true
for all instability regions observed numerically (around 2ωn, ωn,
2
3 ωn, and 1
2 ωn).
0 0.5 1 1.5 2 2.5 3
0
0.02
0.04
0.06
0.08
0.10
0.12
ε
ω
nω
pω
FIGURE 2: Frequency spectrum of the stable transient response
with ζ = 0.01, α = 0.80, and ωe
ωn
= 1.60. The arrows indicate the
direction that the peaks are moving as the parametric excitation
frequency is increased.
Previously, Figure 1 showed that the primary response peak
ωp is not stationary along the frequency axis for all parametric
excitation frequencies ωe. We have noted that it is decreasing
in frequency in Figure 2 as the system approaches an instability
during a speed sweep. Figure 3 shows the frequency of the pri-
mary response peak
ωp
ωn
as a function of the parametric excitation
frequency ωe
ωn
obtained throuh a frequency sweep for three dif-
ferent fluctuation stiffness amplitudes, α = 0.5, 0.65, and 0.80.
The unstable regions are highlighded for the middle value of α.
These unstable regions are characterized by a steady linear in-
crease in the frequency of the primary respose peak ωp. They
are preceeded by a sharp drop in primary response frequency
and followed by a high primary response peak that settles back
down quickly. This asymptotic behavior is noted at 2ωp, ωp, and
3
2 ωp
1. Thus, instability regions occur when a secondary peak ap-
proaches the primary peak ωp and when the primary peaky drops
rapidly on increaseing parametric excitation frequency (or rises
rapidly on decreasing excitation frequency).
0.5 1.0 1.5 2.0 2.5 3.0
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
Parametric excitation frequency (ω /ω )e n
Frequencyofprimaryresponsepeak(ω/ω)np
FIGURE 3: Location of the primary response peak ωp as a func-
tion of the parameteric excitation frequency ωe with ζ = 0.01
and (dotted red) α = 0.50, (solid black) α = 0.65, and (dash
blue) α = 0.80. Unstable regions for α = 0.65 are highlighted
gray.
Figure 4 illustrates more thoroughly the frequency domain
behavior at parametric excitation speeds near and approaching
an instability region. As speed increases through regions of sta-
bility (top two plots), the primary response peak shifts to lower
frequencies. At the same time, a unique secondary peak below
the primary peak moves into higher frequencies. The secondary
peak eventually converges with the primary peak (middle plot).
This convergence corresponds to an instability region character-
ized by unbounded response in the time domain. During instabil-
ity, both peaks increase in frequency. As ωe
ωn
moves past the in-
stability region, the secondary peak continues to shift into higher
1Further instability regions around 2
n ωn (n is an integer) [7] are noted with
decreased damping.
3 Copyright c 2015 by ASME
4. frequencies while the primary peak shifts back down to lower
frequencies. The primary peak will eventually converge with
another secondary peak. This sequence of events repeats until
ωe
ωn
> 2.
CONCLUSIONS
Parmateric stability is a primary concern in paramtrically ex-
cited systems. Though analytical means are capable of deter-
mining system stability, this analaysis shows that the frequency
spectrum of a transient response shows indicators of closeness
to instability regions. The frequency domain shows a primary
response peak near the natural frequency. This frequency peak
is not stationary along the frequency axis. It moves toward
low/high values when the excitation frequency approaches an
instability region with increasing/decreasing frequency, respec-
tively. In addition, there are secondary peaks that approach the
primary peak and coalesce with it as the system enters an in-
stability region. After the instability, the primary resonance has
jumped up to a higher frequency value. This behavior indicates
that the spectum of a system can be used to predict whether a
system is close to an instability region. A system is nearing in-
stability if the primary response peak moves away from the nat-
ural frequency and a secondary peak approaches it. Continued
work will seek to validate these results analytically and experi-
mentally. We expect to discuss this verification at the time of the
presentation.
REFERENCES
[1] Velex, P., and Flamand, L., 1996. “Dynamic response of
planetary trains to mesh parametric excitations”. Journal of
Mechanical Design, 118(1), Mar., pp. 7–14.
[2] Vaishya, M., and Singh, R., 2001. “Sliding friction induced
non-linearity and parametric effects in gear dynamics”. Jour-
nal of Sound and Vibration, 248(4), May, pp. 671–694.
[3] Lin, J., and Parker, R. G., 2002. “Planetary gear parametric
instability caused by mesh stiffness variation”. Journal of
Sound and Vibration, 249(1), Jan., pp. 129–145.
[4] Liu, G., and Parker, R., 2012. “Nonlinear, parametrically
excited dynamics of two-stage spur gear trains with mesh
stiffness fluctuation”. Journal of Mechanical Engineering
Science, 226(C8), pp. 1939–1957.
[5] Han, Q., Wang, J., and Li, Q., 2010. “Frequency response
characteristics of parametrically excited system”. Journal of
Vibration and Acoustics, 132(4), Aug.
[6] Neal, H. L., and Nayfeh, A. H., 1990. “Response of a single-
degree-of-freedom system to a non-stationary principal para-
metric excitation”. International Journal of Non-Linear Me-
chanics, 25(2-3), pp. 275–284.
[7] Nayfeh, A. H., and Mook, D. T., 1995. Nonlinear Oscilla-
tions. Wiley-VCH.
0
0.02
0.04
0.06
ε
0 0.5 1.0 1.5 2.0 2.5 3.0
ω
n
0
0.02
0.04
0.06
ε
0
0.5
1.0
1.5
2.0
2.5 x 105
ε
0
0.1
0.2
0.3
ε
0
0.04
0.08
0.12
0.16
ε
ω
Stable
ω
nω = 1.05e
Stable
ω
nω = 1.03e
Stable
ω
nω = 0.875e
Stable
ω
nω = 0.865e
Stable
ω
nω = 0.855e
pω
pω
pω
pω
pω
FIGURE 4: Frequency spectrum of the transient response of
the parametrically excited single-degree-of-freedom system with
α = 0.8 and ζ = 0.01 for parametric excitation frequencies
ωe
ωn
= 0.86, ωe
ωn
= 0.87, ωe
ωn
= 0.88, ωe
ωn
= 1.03, ωe
ωn
= 1.05.
4 Copyright c 2015 by ASME