Contromechanics Flashcards
What is a mechatronic system?
A mechatronic system is a system that makes use of the synergetic combination of electronics, mechanics and information technology.
What are the three disciplines we use to analyze and define a mechatronic system?
Mechatronics is the interdisciplinary approach to
analyse systems with Mechanical, Electrical and
Information elements using one „language“ to
describe all relevant phenomena.
Give examples for mechatronic systems.
Digital camera, car, robot, production lines, doorbell, …
What is the purpose of the energy-concept within mechatronics?
The energy-concept describes the systems as a combination of energy-storage elements and energy-dissipation elements. The concept can be applied to either mechanics (spring/damper) or electronics (L, C / R).
What are the advantages of the energy concept compared to the classical approach to analyze
mechatronic systems?
The same concept can be used for the different disciplines and is therefore easier to implement for systems that make use of electronics and mechanics. The energy concept also enables us to use generalized coordinates.
The advantage compared to the Newtonian approach is that the equations using the energy approach always have the same form independently of the choice of coordinates. Langrangian methods provide a much elegant and direct way of solving complicated systems.
Sketch a dynamically controlled spring mass system and derive the equations of motion.
m * x’’= F(t) - k*x - d * x’
Sketch an electrical oscillator and derive the governing equations.
V = L q ̈+Rq ̇+1/c *q
q=∫i dt
Compare the equations of motion of a single mass oscillator with the governing equations of an
electric oscillator.
F↔V q↔x L↔m R↔d c↔1/k
How can the physical parameters of an electric oscillator be interpreted from a mechanical
standpoint?
The kinetic energy is stored in the inductance, potential energy is stored in the capacitor and the energy is dissipated in the resistor
Write down the definition of mechanical work.
W = ∫_A^B (F ds ⃗)
F: impressed forces (eingebrachte Kräfte, keine Reaktionskräfte)
For which type of forces are we able to formulate potential energy?
For conservative/restoring forces.
What is the potential energy stored in an elastic spring?
E_pot = 1/2 c x^2
What is the definition of kinetic energy of a mass point m travelling with speed v?
E_kin = 1/2 m * v^2
Give examples for dissipative elements in mechatronic systems.
Damper, resistor
What is energy conservation?
Kinetic Energy is transformed into potential energy and vise versa, while no energy is lost.
E_pot+E_kin=const
What is energy flow?
Kinetic Energy is transformed into potential energy and vise versa.
Explain the principle of virtual work.
The work of a force on a particle along a virtual displacement is known as the virtual work.
Characterize the equilibrium state of mechatronic systems using the principle of virtual work.
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What is collocation of actuator and sensor in a mechatronic system?
The sensor must be collocated to the actuator without being separated by an energy storage element such as an elastic beam, a transmission line or an electromagnetic transformer.
Which problems are we faced with if sensor and actuator are not collocated?
With dislocated sensor, there is a risk of unstable beam-motion at overcritical frequencies when the position is PD-controlled
Give an example of a collocated and a not collocated actuator-sensor configuration.
Force and sensor are located at the same point on beam.
Force and sensor are NOT located at the same point on beam.
What is the purpose of calculus of variations?
The general purpose of calculus of variations is to solve optimization problems.
Calculus of variation seeks to find the path, curve or surface for which a given function has a stationary value. In physical problems, this is usually a minimum or a maximum.
Formulate a variational problem.
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Given a proper formulated variational problem write down the corresponding differential
equation.
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What is the name of this differential equation?
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Derive the equations of motion of a mechanical oscillator using the Euler Lagrange equation.
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Derive the governing equations of an electric oscillator using the Euler Lagrange equation.
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What are generalized coordinates?
The minimum number of independent coordinates that describe the system completely.
Give examples of coordinates describing the position of a mass points and compare to
generalized coordinates. Explain Hamilton’s Principle.
x,y: standard coordinates
ϕ: generalized coordinate
What is the purpose of Lagrange multipliers?
The method is a strategy for finding local minima and maxima of a function subject to equality constraints.
Derive the equation of motion of a pendulum using cartesian coordinates and the Lagrange multipliers.
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What is the definition of potential energy stored in an elastic beam?
U=1/2 EI∫w^’‘2 dx
What is the definition of kinetic energy of an elastic beam?
T=1/2 ρ A ∫ w ̇^2 dx
How can we derive the equations of motion of an elastic beam with distributed mass?
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How can we discretize the equations of motion of a one dimensional continuum like the elastic beam?
Ritz-Galerkin
What are admissible shape functions for the process of discretization?
Shape functions: 𝑊𝑛 𝑥 = 𝛼𝑛 sin (𝛽𝑛𝑥),
𝛽𝑛 =𝑛𝜋/𝑙
𝑊𝑛 𝑥 = 𝛼𝑛 sin (𝑛𝜋/𝑙 *𝑥),
What are natural boundary conditions?
Forces and moments
What are geometric and dynamic boundary conditions?
Essential boundary conditions
What is the Rayleigh-Ritz method?
The Rayleigh-Ritz principle is a principle to determine the smallest eigenvalue of an eigenvalue problem
Explain the Maximum Principle of Pontrjagin.
Is a principle that can be used to bring a system from one state to another in an optimally controlled way.
Explain how to move a mass point m in minimum time from a point A in a plane to a point B on the
same plane. What is a phase- diagram?
Using the greatest possible amount of force and power.
What is bang-bang control?
Bang-bang control refers to a feedback controller, that switches between two states with maximum power outputs.
Explain why minimum time problems need a bang-bang type of control.
Because it is the fastest control to bring a system to the desired state.
What are the drawbacks of using the Pontrjagin concept without modifications in real life systems?
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