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Question 2: Coupled Systems with Friction Consider a system with two DoFs such that the Lagrangia...
Classical Mechanics problem: Consider the two coupled pendulums shown in the figure below. Each of the...
Classical Mechanics problem:
Consider the two coupled pendulums shown in the figure below. Each of the pen- dulums has a length L and the spring constant is k. The pendulums' position can be specified by the angles ¢\ and ø2. The relaxed length of the spring is such that the equi librium position of the pendulums is at ¢2 = 0 with the two pendulums vertical a.) Find the lagrangian L of this system. You can assume the angular deflections...
I. COUPLED OSCILLATIONS Consider the system below with two degrees of freedom (neglect gravitation). Denote the...
I. COUPLED OSCILLATIONS Consider the system below with two degrees of freedom (neglect gravitation). Denote the displacement of each of the particles with respect to equilibrium by óy (t), i= 1,2. 1. Find the Lagrangian describing the system 2. Write down the coupled equations of motion for óyn (t) and by2(t) 3. Find the 2 x 2 matrices Ť and V and solve the normal mode equation for w: det(V-2T) 0. 4. Compute the form of the eigenvectors (normal modes),...
Problem 3 (10 points). Consider the weakly coupled mechanical system shown in figure 2. Let are:...
Problem 3 (10 points). Consider the weakly coupled mechanical system shown in figure 2. Let are: k be the stiffness of the spring and mi-m2- m. Given that the initial conditions 1(0)0 6,(0)-A Oz(0)-0 02(0)0 I. Compute the complete solution of the system linearized around θ1 θ2 0 2. Given the numerical values in the following table, plot θ1(t) and θ2(t) on the same figure for 0 << 100s. Give a physical interpretation of what is happening Parameter Numerical Value...
We consider 2 coupled harmonic oscillators, as shown in the diagram below. The mass m1 is...
We consider 2 coupled harmonic oscillators, as shown in the
diagram below.
The mass m1 is subjected to an external force F (t).
1. Construct the system of differential equations whose unknowns
are the displacements x1 (t) and x2 (t) of each of the 2 masses. (5
points).
2. Solve x1(t) and x2(t) in the case where
m1 = 1kg; m2 = 2kg; k = 1 N / m; F(t) = 0 and
x1(0) = 0; ?1′(0) = 0; x2(0)...
We consider 2 coupled harmonic oscillators, as shown in the diagram below The mass m1 is...
We consider 2 coupled harmonic oscillators, as shown in the
diagram below
The mass m1 is subjected to an external force F(t).
1) Construct the system of differential equations whose unknowns
are the displacements x1 (t) and x2 (t) of each of the 2
masses.
2) Solve x1 (t) and x2 (t) in the case where m1 = 1kg; m2 = 2kg;
k = 1 N / m; F (t) = 0 and x1 (0) = 0; ?1′ (0) =...
Frictionless plane M 1.) Consider the coupled system shown at the right. The mass M is...
Frictionless plane M 1.) Consider the coupled system shown at the right. The mass M is free to slide on a frictionless surface and is connected to the wall with a spring of spring constant k. Mass M2 is 2000 attached to My with taut rope of length (it acts as a pendulum). The vertical line shows the equilibrium position when the spring is un- stretched (r = 0). The coordinates 21 and 12 denote the positions of the two...
DIFFÉRENTIEL EQUATIONS We consider 2 coupled harmonic oscillators, as shown in the diagram below. The mass...
DIFFÉRENTIEL EQUATIONS
We consider 2 coupled harmonic oscillators, as shown in the
diagram below.
The mass m1 is subjected to an external force F (t).
1. Construct the system of differential equations whose unknowns
are the displacements x1 (t) and x2 (t) of each of the 2 masses
(DIFFERENTIEL EQUATIONS).
2. Solve x1 (t) and x2 (t) in the case where m1 = 1kg; m2 = 2kg;
k = 1 N / m; F (t) = 0 and x1 (0)...
4. Consider the pendulum with friction modeled by the second order ODE: where θ is the angle the ...
4. please help with both parts a and b
4. Consider the pendulum with friction modeled by the second order ODE: where θ is the angle the pendulum makes with the vertical axis, α is a friction coefficient and w is the pendulum natural frequency. (a) Turn (4) into a first order system. (b) Use Euler method to find an approximation to the solution in [0,5] with initial conditions θ(0)-1 and θ'(0)-0. Take α-0.2 and w-2. Verify the expected order...
4. (30%) Consider a 2-dof system with 8 -2 1 0 [m] = 0 4 [k]...
4. (30%) Consider a 2-dof system with 8 -2 1 0 [m] = 0 4 [k] = -2 2 (a) Find the natural frequencies and mode shapes of the system. (b) Find the uncoupled modal equations. (c) If the system has proportional damping and it is known that the damping ratios are both 0.05 for the 2 modes, find the damping matrix of the system.
2. In many mechanical positioning systems there is flexibility between one part of the system and...
2. In many mechanical positioning systems there is flexibility between one part of the system and another. The figure below depicts such a situation, where a force u(t) is applied to the mass M, and another mass m is connected to it. The coupling between the objects is often modeled by a spring constant k with a damping coefficient b, although the actual situation is usually much more complicated than this. y(t) m M ut) no friction no friction a)...
Classical Mechanics problem:
Consider the two coupled pendulums shown in the figure below. Each of the pen- dulums has a length L and the spring constant is k. The pendulums' position can be specified by the angles ¢\ and ø2. The relaxed length of the spring is such that the equi librium position of the pendulums is at ¢2 = 0 with the two pendulums vertical a.) Find the lagrangian L of this system. You can assume the angular deflections...
I. COUPLED OSCILLATIONS Consider the system below with two degrees of freedom (neglect gravitation). Denote the displacement of each of the particles with respect to equilibrium by óy (t), i= 1,2. 1. Find the Lagrangian describing the system 2. Write down the coupled equations of motion for óyn (t) and by2(t) 3. Find the 2 x 2 matrices Ť and V and solve the normal mode equation for w: det(V-2T) 0. 4. Compute the form of the eigenvectors (normal modes),...
Problem 3 (10 points). Consider the weakly coupled mechanical system shown in figure 2. Let are: k be the stiffness of the spring and mi-m2- m. Given that the initial conditions 1(0)0 6,(0)-A Oz(0)-0 02(0)0 I. Compute the complete solution of the system linearized around θ1 θ2 0 2. Given the numerical values in the following table, plot θ1(t) and θ2(t) on the same figure for 0 << 100s. Give a physical interpretation of what is happening Parameter Numerical Value...
We consider 2 coupled harmonic oscillators, as shown in the
diagram below.
The mass m1 is subjected to an external force F (t).
1. Construct the system of differential equations whose unknowns
are the displacements x1 (t) and x2 (t) of each of the 2 masses. (5
points).
2. Solve x1(t) and x2(t) in the case where
m1 = 1kg; m2 = 2kg; k = 1 N / m; F(t) = 0 and
x1(0) = 0; ?1′(0) = 0; x2(0)...
We consider 2 coupled harmonic oscillators, as shown in the
diagram below
The mass m1 is subjected to an external force F(t).
1) Construct the system of differential equations whose unknowns
are the displacements x1 (t) and x2 (t) of each of the 2
masses.
2) Solve x1 (t) and x2 (t) in the case where m1 = 1kg; m2 = 2kg;
k = 1 N / m; F (t) = 0 and x1 (0) = 0; ?1′ (0) =...
Frictionless plane M 1.) Consider the coupled system shown at the right. The mass M is free to slide on a frictionless surface and is connected to the wall with a spring of spring constant k. Mass M2 is 2000 attached to My with taut rope of length (it acts as a pendulum). The vertical line shows the equilibrium position when the spring is un- stretched (r = 0). The coordinates 21 and 12 denote the positions of the two...
DIFFÉRENTIEL EQUATIONS
We consider 2 coupled harmonic oscillators, as shown in the
diagram below.
The mass m1 is subjected to an external force F (t).
1. Construct the system of differential equations whose unknowns
are the displacements x1 (t) and x2 (t) of each of the 2 masses
(DIFFERENTIEL EQUATIONS).
2. Solve x1 (t) and x2 (t) in the case where m1 = 1kg; m2 = 2kg;
k = 1 N / m; F (t) = 0 and x1 (0)...
4. please help with both parts a and b
4. Consider the pendulum with friction modeled by the second order ODE: where θ is the angle the pendulum makes with the vertical axis, α is a friction coefficient and w is the pendulum natural frequency. (a) Turn (4) into a first order system. (b) Use Euler method to find an approximation to the solution in [0,5] with initial conditions θ(0)-1 and θ'(0)-0. Take α-0.2 and w-2. Verify the expected order...
4. (30%) Consider a 2-dof system with 8 -2 1 0 [m] = 0 4 [k] = -2 2 (a) Find the natural frequencies and mode shapes of the system. (b) Find the uncoupled modal equations. (c) If the system has proportional damping and it is known that the damping ratios are both 0.05 for the 2 modes, find the damping matrix of the system.
2. In many mechanical positioning systems there is flexibility between one part of the system and another. The figure below depicts such a situation, where a force u(t) is applied to the mass M, and another mass m is connected to it. The coupling between the objects is often modeled by a spring constant k with a damping coefficient b, although the actual situation is usually much more complicated than this. y(t) m M ut) no friction no friction a)...
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