# Beam Dynamics Issues for 1 TeV Linear Collider

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The elastic modulus of concrete is E = 3000 ksi. 45 kips/ft. 2 The longitudinal stiffness of the bridge is computed assuming the bridge deck to displace rigidly as shown in Fig. 3c. Each column of a bent behaves as a clamped–clamped column. 6 MASS–SPRING–DAMPER SYSTEM We have introduced the SDF system by idealizing a one-story structure (Fig. 1a), an approach that should appeal to structural engineering students. However, the classic SDF system is the mass–spring–damper system of Fig. 1a. The dynamics of this system is developed in textbooks on mechanical vibration and elementary physics.

22 Equations of Motion, Problem Statement, and Solution Methods Chap. 5 Derive the equation governing the free motion of a simple pendulum (Fig. 5a), which consists of a point mass m suspended by a light string of length L. 5 (a) Simple pendulum; (b) free-body diagram. 5a shows the displaced position of the pendulum defined by the angle θ measured from the vertical position, and Fig. 5b shows the free-body diagram of the mass. The forces acting are the weight mg, tension T in the string, and D’Alembert’s fictitious inertia force f I = m L θ¨ .

5b shows the free-body diagram of the mass. The forces acting are the weight mg, tension T in the string, and D’Alembert’s fictitious inertia force f I = m L θ¨ . Equilibrium of the moments of forces about O gives m L 2 θ¨ + mgLsin θ = 0 (a) This is a nonlinear differential equation governing θ. For small rotations, sin θ θ and the equation of motion [Eq. 6 The system of Fig. 6 consists of a weight w attached to a rigid massless bar of length L joined to its support by a rotational spring of stiffness k.