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IV. Spring-Mass System Application - Consider the system of two masses and three springs as shown...
Three identical masses are coupled together by four identical springs. The position of the left-most mass...
Three identical masses are coupled together by four identical springs. The position of the left-most mass is 21, the position of the next mass is ry and the final mass is located at position Z3, as shown in the diagram below. பண்டண்டண்டண் | m m m X2 Using Newton's second law, we find the following equations govern the motion of these three masses. _m = =-kz - ke(z) – 22) m" --k(x2 – £1 ) - k:(:2 – £3) m...
5. 10 points Two ideal massless springs with spring constant k are connected to two masses...
5. 10 points Two ideal massless springs with spring constant k are connected to two masses that hang vertically as shown in the figure. The top one has mass 3m and the bottom one has mass 2m. The system is only able to oscillate in the vertical direction. a) Determine the equations of motion. (4 pts) b) Find the frequencies of the normal modes of this system for small vertical dis- placements. (4 pts) c) Describe the relative motion and...
3. Consider the spring - mass system shown below, consisting of two masses mi and ma sus- pended ...
3. Consider the spring - mass system shown below, consisting of two masses mi and ma sus- pended from springs with spring constants ki and k, respectively. Assume that there is no damping in the system. a) Show that the displacements z1 and 2 of the masses from their respective equilibrium positions satisfy the differential equations b) Use the above resuit to show that the spring-mass system satisfies the following fourth order differential equation. and ) Find the general solution...
3. Consider the spring - mass system shown below, consisting of two masses mi and m2 sus- pended ...
3. Consider the spring - mass system shown below, consisting of two masses mi and m2 sus- pended from springs with spring constants ki and k2, respectively. Assume that there is no damping in the system. a) Show that the displacements ai and r2 of the masses from their respective equilibrium positions satisfy the differential equations b) Use the above result to show that the spring-mass system satisfies the following fourth order differential equation and c) Find the general solution...
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....
Consider the system of two equal masses M joined together by three identical springs of spring co...
Consider the system of two equal masses M joined together by three identical springs of spring constant k. *2 x1 As shown in the figure, assume the left mass has been displaced a from its equilibrium position, and the right mass has been displaced distance a distance T2 from its equilibrium position. In terms of ri and z2 i. How much has the left spring been stretched/compressed from equilibrium? ii. How much has the middle spring been stretched/compressed from equilib-...
5. (11 pts) Consider two masses connected by springs (as shown belowe), assume the movement is fr...
5. (11 pts) Consider two masses connected by springs (as shown belowe), assume the movement is frictionles, and that z(t) and y(t) represent the position of mass l and mass 2 respectively. System of masses and springs (a)If m.-5, rni-4 kĩ = 10,and ka= 20, findthe matrix Aso that thesystem is rnodeled by F]-4 (b) The matrix A has eigenvalses -1 and -10, with respective egenvectorsand solution for the position r(t) v(t)
5. (11 pts) Consider two masses connected by...
6.(16) Consider the spring-mass system shown, consisting of two unit masses m, and my suspended from...
6.(16) Consider the spring-mass system shown, consisting of two unit masses m, and my suspended from springs with constants k, and ky, respectively. Assuming that there is no damping in the system, the displacement y(t) of the bottom mass m, from its equilibrium positions satisfies the 4-order equation (4) y2 + k + k)y + k_k2yz = e-2, where f(t) = e-2 is an outside force driving the motion of m. If a 24 N weight would stretch the top...
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...
Solve the following problems: Problem 1: masses&springs Two masses mand m2 connected by a spring of...
Solve the following problems:
Problem 1: masses&springs Two masses mand m2 connected by a spring of elastic constant k slide on a frictionless inclined plane under the effect of gravity. Let a be the angle between the the x axis and the inclined plane, r the distance between the two masses, l the position of the first mass with respect to the top of the plane (see figure). Considering the top of the plane to be the zero for potential...
Three identical masses are coupled together by four identical springs. The position of the left-most mass is 21, the position of the next mass is ry and the final mass is located at position Z3, as shown in the diagram below. பண்டண்டண்டண் | m m m X2 Using Newton's second law, we find the following equations govern the motion of these three masses. _m = =-kz - ke(z) – 22) m" --k(x2 – £1 ) - k:(:2 – £3) m...
5. 10 points Two ideal massless springs with spring constant k are connected to two masses that hang vertically as shown in the figure. The top one has mass 3m and the bottom one has mass 2m. The system is only able to oscillate in the vertical direction. a) Determine the equations of motion. (4 pts) b) Find the frequencies of the normal modes of this system for small vertical dis- placements. (4 pts) c) Describe the relative motion and...
3. Consider the spring - mass system shown below, consisting of two masses mi and ma sus- pended from springs with spring constants ki and k, respectively. Assume that there is no damping in the system. a) Show that the displacements z1 and 2 of the masses from their respective equilibrium positions satisfy the differential equations b) Use the above resuit to show that the spring-mass system satisfies the following fourth order differential equation. and ) Find the general solution...
3. Consider the spring - mass system shown below, consisting of two masses mi and m2 sus- pended from springs with spring constants ki and k2, respectively. Assume that there is no damping in the system. a) Show that the displacements ai and r2 of the masses from their respective equilibrium positions satisfy the differential equations b) Use the above result to show that the spring-mass system satisfies the following fourth order differential equation and c) Find the general solution...
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....
Consider the system of two equal masses M joined together by three identical springs of spring constant k. *2 x1 As shown in the figure, assume the left mass has been displaced a from its equilibrium position, and the right mass has been displaced distance a distance T2 from its equilibrium position. In terms of ri and z2 i. How much has the left spring been stretched/compressed from equilibrium? ii. How much has the middle spring been stretched/compressed from equilib-...
5. (11 pts) Consider two masses connected by springs (as shown belowe), assume the movement is frictionles, and that z(t) and y(t) represent the position of mass l and mass 2 respectively. System of masses and springs (a)If m.-5, rni-4 kĩ = 10,and ka= 20, findthe matrix Aso that thesystem is rnodeled by F]-4 (b) The matrix A has eigenvalses -1 and -10, with respective egenvectorsand solution for the position r(t) v(t)
5. (11 pts) Consider two masses connected by...
6.(16) Consider the spring-mass system shown, consisting of two unit masses m, and my suspended from springs with constants k, and ky, respectively. Assuming that there is no damping in the system, the displacement y(t) of the bottom mass m, from its equilibrium positions satisfies the 4-order equation (4) y2 + k + k)y + k_k2yz = e-2, where f(t) = e-2 is an outside force driving the motion of m. If a 24 N weight would stretch the top...
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...
Solve the following problems:
Problem 1: masses&springs Two masses mand m2 connected by a spring of elastic constant k slide on a frictionless inclined plane under the effect of gravity. Let a be the angle between the the x axis and the inclined plane, r the distance between the two masses, l the position of the first mass with respect to the top of the plane (see figure). Considering the top of the plane to be the zero for potential...
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