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) = 0; x2 (0) = 0; ?2′ (0) = 0
3) Solve x1 (t) and x2 (t) in the case where m1 = 1kg; m2 = 2kg; k = 1 N / m; F (t) = cos(t) and x1 (0) = 0; ?1′ (0) = 0; x2 (0) = 0; ?2′ (0) = 0
Homework Answers
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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...
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)...
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)...
6. Consider two coupled oscillators of mass mi and m2 that are attached by springs and are unstre...
6. Consider two coupled oscillators of mass mi and m2 that are attached by springs and are unstretched when x,-12-0. The damping force viscous dampers. The springs is proportional to the speed of the displacement and acts in the direction opposite the motion, Fd--cv T1 k1 k2 m1 m2 C1 C2 C3 a) Find the equations of motion in and i2 b) Since the equations are coupled, write them in matrix form: ME+ CE+ K 0
6. Consider two coupled...
M1 m2 Figure 1: 2dof 1. Consider the system above. Derive the equation of motion and calculate th...
m1 m2 Figure 1: 2dof 1. Consider the system above. Derive the equation of motion and calculate the mass and stiffness matrices Note that setting k30 in your solution should result in the stiffness matrix given by Eq. (4.9). a. Calculate the characteristic equation from problem 4.1 for the case m1-9 kg m2-1 kg ki-24 N/m 2 3 N/m k 3 N/m and solve for the system's natural frequencies. b. Calculate the eigenvectors u1 and u2. c. Calculate 띠(t) and...
We consider here, the two masses m1 and m2 connected this time by springs of stiffnesses...
We consider here, the two masses m1 and m2 connected this time
by springs of stiffnesses k1, k2 and k3 as shown in the figure
below. We denote x1 (t) and x2 (t) as the movement of each of the 2
masses relative to its position of equilibrium static.
1) Prove that the differential equation whose unknown is the displacement is written in the following form:
2) Deduce the second differential equation whose unknown is the
displacement
3) Determine the...
Differentiel equations We consider here, the two masses m1 and m2 connected this time by springs...
Differentiel equations
We consider here, the two masses m1 and m2 connected this time
by springs of stiffnesses k1, k2 and k3 as indicated in the figure
below. We denote by x1 (t) and x2 (t) the movement of each of the 2
masses relative to its static equilibrium position.
1. Prove that the differential equation whose unknown is the
displacement x1 (t) is written in the following form:
2. Deduce the second differential equation whose unknown is the
displacement...
We consider here, the two masses m1 and m2 connected this time by springs of stiffnesses...
We consider here, the two masses m1 and m2
connected this time by springs of stiffnesses k1,
k2 and k3 as shown in the figure below. We
denote by x1(t) and x2(t) the movement of
each of the 2 masses relative to its position of equilibrium
static.
1. Prove that the differential equation whose unknown is the
displacement x1(t) is written in the following form: (3
points)
2. Deduce the second differential equation whose unknown is the
displacement x2(t) (3...
Here we consider the two masses m1 and m2 connected this time by springs of stiffnesses...
Here we consider the two masses m1 and m2 connected this time by
springs of stiffnesses k1, k2 and k3 as shown in the figure below.
The movement of each of the 2 masses relative to its position of
static equilibrium is designated by x1(t) and x2(t).
1. Demonstrate that the differential equation whose unknown is
the displacement x1(t) is written as follows:
2. Determine the second differential equation whose unknown is
the displacement x2(t).
3. Determine the free oscillatory...
(6) Here is the Hamiltonian for two coupled harmonic oscillators oriented along ti same axis: 22...
(6) Here is the Hamiltonian for two coupled harmonic oscillators oriented along ti same axis: 22 22 2mozz + 3.2) + -k(zỉ + 2) + Ariza, (a) What are the energy levels for the two oscillators when there is no coupling, or i = 0 (b) Now include the coupling. What change of variables can be made to solve this syste exactly? (C) What are the energy levels for the system? (d) How would one construct the raising and lowering...
4. Two masses mi and m2 are connected to three springs of negligible mass having spring constants...
4. Two masses mi and m2 are connected to three springs of negligible mass having spring constants k1, k2 and k3, respectively. x2=0 Il k, Let xi and x2 represent The motion of the equations: displacements of masses mi and m2 from their equilibrium positions . coupled system is represented by the system of second-order differential d2x dt2 d2x2 Using Laplace transform to solve the system when k1 1 and x1(0) = 0, xi (0)--1 , x2(0) = 0, x(0)-1....
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)...
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)...
6. Consider two coupled oscillators of mass mi and m2 that are attached by springs and are unstretched when x,-12-0. The damping force viscous dampers. The springs is proportional to the speed of the displacement and acts in the direction opposite the motion, Fd--cv T1 k1 k2 m1 m2 C1 C2 C3 a) Find the equations of motion in and i2 b) Since the equations are coupled, write them in matrix form: ME+ CE+ K 0
6. Consider two coupled...
m1 m2 Figure 1: 2dof 1. Consider the system above. Derive the equation of motion and calculate the mass and stiffness matrices Note that setting k30 in your solution should result in the stiffness matrix given by Eq. (4.9). a. Calculate the characteristic equation from problem 4.1 for the case m1-9 kg m2-1 kg ki-24 N/m 2 3 N/m k 3 N/m and solve for the system's natural frequencies. b. Calculate the eigenvectors u1 and u2. c. Calculate 띠(t) and...
We consider here, the two masses m1 and m2 connected this time
by springs of stiffnesses k1, k2 and k3 as shown in the figure
below. We denote x1 (t) and x2 (t) as the movement of each of the 2
masses relative to its position of equilibrium static.
1) Prove that the differential equation whose unknown is the displacement is written in the following form:
2) Deduce the second differential equation whose unknown is the
displacement
3) Determine the...
Differentiel equations
We consider here, the two masses m1 and m2 connected this time
by springs of stiffnesses k1, k2 and k3 as indicated in the figure
below. We denote by x1 (t) and x2 (t) the movement of each of the 2
masses relative to its static equilibrium position.
1. Prove that the differential equation whose unknown is the
displacement x1 (t) is written in the following form:
2. Deduce the second differential equation whose unknown is the
displacement...
We consider here, the two masses m1 and m2
connected this time by springs of stiffnesses k1,
k2 and k3 as shown in the figure below. We
denote by x1(t) and x2(t) the movement of
each of the 2 masses relative to its position of equilibrium
static.
1. Prove that the differential equation whose unknown is the
displacement x1(t) is written in the following form: (3
points)
2. Deduce the second differential equation whose unknown is the
displacement x2(t) (3...
Here we consider the two masses m1 and m2 connected this time by
springs of stiffnesses k1, k2 and k3 as shown in the figure below.
The movement of each of the 2 masses relative to its position of
static equilibrium is designated by x1(t) and x2(t).
1. Demonstrate that the differential equation whose unknown is
the displacement x1(t) is written as follows:
2. Determine the second differential equation whose unknown is
the displacement x2(t).
3. Determine the free oscillatory...
(6) Here is the Hamiltonian for two coupled harmonic oscillators oriented along ti same axis: 22 22 2mozz + 3.2) + -k(zỉ + 2) + Ariza, (a) What are the energy levels for the two oscillators when there is no coupling, or i = 0 (b) Now include the coupling. What change of variables can be made to solve this syste exactly? (C) What are the energy levels for the system? (d) How would one construct the raising and lowering...
4. Two masses mi and m2 are connected to three springs of negligible mass having spring constants k1, k2 and k3, respectively. x2=0 Il k, Let xi and x2 represent The motion of the equations: displacements of masses mi and m2 from their equilibrium positions . coupled system is represented by the system of second-order differential d2x dt2 d2x2 Using Laplace transform to solve the system when k1 1 and x1(0) = 0, xi (0)--1 , x2(0) = 0, x(0)-1....
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