Question

EXAMPLE Two simple pendulums of the same length are connected by a light spring with stiffness $k$ as shown. For this system, determine (a) the equations of motion (b) the natural frequencies (c) the mode shapes (d) the total response of the system (for $t > 0$) if the initial conditions at time $t = 0$ are $\theta_1 = \phi_0$ $\dot{\theta}_1 = 0$ $\theta_2 = 0$ $\dot{\theta}_2 = 0$ For parts (b)-(d) assume that $m_1 = m_2 = m.$

          EXAMPLE
Two simple pendulums of the same length are connected by a light spring with
stiffness $k$ as shown.


For this system, determine
(a) the equations of motion
(b) the natural frequencies
(c) the mode shapes
(d) the total response of the system (for $t > 0$) if the initial conditions at time $t = 0$ are
$\theta_1 = \phi_0$
$\dot{\theta}_1 = 0$
$\theta_2 = 0$
$\dot{\theta}_2 = 0$
For parts (b)-(d) assume that $m_1 = m_2 = m.$

EXAMPLE
Two simple pendulums of the same length are connected by a light spring with
stiffness k as shown.

For this system, determine
(a) the equations of motion
(b) the natural frequencies
(c) the mode shapes
(d) the total response of the system (for t > 0) if the initial conditions at time t = 0 are
θ1 = ϕ0
θ̇1 = 0
θ2 = 0
θ̇2 = 0
For parts (b)-(d) assume that m1 = m2 = m.

Added by Yvonne M.

Mathematical Methods in the Physical Sciences

Mary L. Boas 2nd Edition

Chapter 10

Instant Answer

Solved by Expert Mkhitar Hobosyan

Step 1

Let's denote the angles of the pendulums as θ1 and θ2, and their angular velocities as θ1' and θ2'. The forces acting on each pendulum are the gravitational force and the spring force. For pendulum 1: - Gravitational force: m1 * g * sin(θ1) - Spring force: k * Show more…

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Two simple pendulums of the same length are connected by a light spring with stiffness as shown. For this system, determine (a) the equations of motion, (b) the natural frequencies, (c) the mode shapes, and (d) the total response of the system (for t > 0) if the initial conditions at time t = 0 are: Î¸1 = Î¸2 = 0 Î¸1' = Î¸2' = 0 For parts (b) to (d), assume that m1 = m2 = m.