Me330 lecture5
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This document discusses modeling mechanical systems for control systems. It covers: 1) Newton's second law governs mechanical systems and results in equations of motion describing dynamical systems. These equations can be represented using block diagrams and Laplace transforms. 2) Modeling involves determining the equation of motion using free body diagrams and summing the forces. Mechanical components like springs, dampers and masses have characteristic force-velocity, force-displacement and impedance relationships. 3) Systems with multiple degrees of freedom require equations of motion equal to the number of independent motions. Transfer functions can be derived from the Laplace transform of the equations of motion.
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This document discusses multi degree of freedom systems and vehicle dynamics. It contains the following key points:
- Multi degree of freedom systems require two or more coordinates to describe motion and have multiple natural frequencies and vibration modes. Numerical analysis is required for systems with more than two degrees of freedom.
- The equations of motion for these systems can be written in matrix form using mass and stiffness matrices. Assuming harmonic motion, the equations can be reduced to an eigenvalue problem.
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This document discusses Betti's and Maxwell's laws of reciprocal deflections for linearly elastic structures. Betti's theorem states that the work done by a set of external forces Pm acting through displacements mn produced by another set of forces Pn is equal to the work done by Pn acting through displacements nm produced by Pm. Maxwell's law of reciprocal deflection states that the deflection of point n due to a force P at point m is numerically equal to the deflection of point m due to the same force P applied at point n. The total external work done on a structure by two sets of forces Pm and Pn applied in different sequences must be the same according to the principleLateral or transverse vibration of thin beam
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1. Betti's theorem from 1872 states that for linear elastic structures under two sets of forces P and Q, the work done by P through displacements caused by Q is equal to the work done by Q through displacements caused by P.
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3) Static equilibrium equations must be satisfied for coplanar forces. Systems can be determinate, allowing determination of specific unknowns, or indeterminate.Betti
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1. Betti's theorem from 1872 states that for linear elastic structures under two sets of forces P and Q, the work done by P through displacements caused by Q is equal to the work done by Q through displacements caused by P.
2. An example is given of a beam with two points where a force P is applied to point 1, displacing point 2, and then a force Q is applied to point 2. Betti's theorem says the work is equal in both cases.
3. The Maxwell-Betti reciprocal theorem extends this idea and is illustrated using a simply supported beam with two load points.Single degree of freedom system free vibration part -i and ii
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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.III - resultant of non-concurrent forces
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This document discusses calculating the resultant force of non-concurrent forces. It provides the equations to calculate the x and y components of the resultant force and the moment. It then provides examples of calculating the resultant force and point of application for different force systems acting on structures. Continuous Charge Distributions, Example
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The document discusses a system of a uniformly charged circular ring and a straight wire with a linear charge density, deriving the electric forces exerted between them. It provides detailed calculations for the electric field at specific points and integrates to find the total forces involved. Ultimately, it concludes that the resulting electric force behaves according to Newtonian principles.(S.h.m & waves).(reflection & refrection).(interferenc) prove s & important ...
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1. The document discusses several physics concepts including simple harmonic motion, waves, reflection, refraction, and interference.
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1) The document discusses ordinary differential equations that model free oscillations of a spring-mass system and electric circuits.
2) It derives the differential equation for simple harmonic motion of a mass attached to a spring as well as cases with damping.
3) Kirchhoff's voltage law is used to derive differential equations for simple R-L and R-C circuits driven by external voltages.
4) As an example, the differential equation for a series R-C circuit is solved analytically to find expressions for charge and current over time.Viewers also liked (8)
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This document provides an overview and logistics for an ME 330 Control Systems course. It introduces the instructor, textbook, grading breakdown, software to be used, and course objectives. Key points are:
- Lectures on Mondays and Wednesdays from 7-8pm, labs on Tuesdays and Thursdays from 5:30-7pm
- The course aims to help students recognize, model, analyze, and design control systems using mathematical tools
- Control systems regulate processes using inputs, outputs, and feedback to handle disturbancesMe330 lecture3
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This document discusses the process of solving differential equations representing dynamical systems using Laplace transforms. It involves 4 main steps: 1) Represent the differential equation in the Laplace domain as a "transfer function", 2) Transform the input function to the Laplace domain, 3) Multiply the transfer function by the input function, 4) Take the inverse Laplace transform to find the time domain response. It also discusses partial fraction expansion, which simplifies functions for taking the inverse Laplace transform using lookup tables. There are three cases for partial fraction expansion depending on the roots of the denominator polynomial.Final j314 slideshow yunker
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1. The document discusses different cases for taking the inverse Laplace transform of a function F(s), depending on the characteristics of the denominator of F(s).
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3. If there are real repeated roots, one multiplies F(s) by the appropriate power and differentiates to find all residues needed for the inverse Laplace transform.
4. If there are complex roots, one solves for the residues by balancing coefficients or factorizing, then combines like terms to find the inverse Laplace transform.Me330 lecture7
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This document discusses control systems and summarizes key concepts from lecture 7:
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2) Electrical and mechanical systems can be combined. Lorentz force and Faraday's law relate current, magnetic fields, and motion in devices like DC servomotors.
3) Transfer functions describe the input-output behavior of electrical, mechanical, and combined electro-mechanical systems. The overall transfer function is derived by substituting Lorentz and Faraday equations into the individual system transfer functions.Me330 lecture2
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1) The document discusses complex numbers and the Laplace transform, which can convert differential equations describing dynamic systems into algebraic equations.
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Me330 lecture5
- 1. ME 330 Control Systems SP 2011 Lecture 5
- 2. Mechanical Systems Newtons 2 nd Law governs mechanical systems, resulting equation of motion describes dynamical system. f(t) From the Laplace transform F(s) X(s) Often seen in block-diagram representation
- 3. Mechanical Systems Modeling Determining the equation of motion Free-Body Diagram Summation of forces Impedance Summation of impedances f(t) m m
- 4. Mechanical Components Table 2.4 Force-velocity, force-displacement, and impedance for springs, viscous dampers, and mass
- 5. Mechanical Components Table 2.5 Torque-angular velocity, torque-angular displacement, and rotational impedance for springs, viscous dampers, and inertia Note: rotational mechanics are analogous to translational mechanics force => torque damper => damper mass => inertia
- 6. Multiple Degrees of Freedom Number of equations of motion required = number of linear independent motions Linear independence : point of motion is allowed to move even if all other points of motion are fixed. Also known as degrees of freedom .
- 7. Multiple Degrees of Freedom Analyze impedances for each degree of freedom. m 1 m 2
- 8. Transfer Function Laplace transform of equations of motion Can solve for any transfer function
- 9. Equations by Inspection For two degree of freedom system, the general form is given by Higher degree of freedom systems Sum of impedances connected to the motion at x 1 Sum of impedances between x 1 and x 2 X 1 (s) X 2 (s) = Sum of applied forces at x 1 Sum of impedances connected to the motion at x 2 Sum of impedances between x 1 and x 2 X 2 (s) X 1 (s) = Sum of applied forces at x 2 +
- 10. Torsion Mechanical Systems Same rules apply as in translational mechanical systems Sum of impedances connected to the motion at 1 Sum of impedances between 1 and 2 1 (s) 2 (s) = Sum of applied forces at 1 Sum of impedances between 1 and 3 3 (s) Sum of impedances connected to the motion at 2 Sum of impedances between 1 and 2 2 (s) 1 (s) = Sum of applied forces at 2 Sum of impedances between 2 and 3 3 (s) + Sum of impedances connected to the motion at 3 Sum of impedances between 1 and 3 3 (s) 1 (s) = Sum of applied forces at 3 Sum of impedances between 2 and 3 2 (s) +
- 11. Torsion Mechanical Systems Summation of impedances
- 12. Next Lectures Derivation of more mechanical system models Derivation of electrical system models