Fundamentals of Finite Element Analysis phần 9

Các điều kiện ban đầu sản xuất một hệ thống của bốn phương trình đại số trong bốn ẩn số A (1), A (2), 1, 2 Giải pháp của các phương trình là không nhỏ, do sự hiện diện 2 2 của các hàm lượng giác. Cho P = A (1) sin 1 và Q = A (2) sin 2, displace2 2 phương trình điều kiện thuận ban đầu trở thành | I Text Hutton Fundamentals of 10. Structural Dynamics Finite Element Analysis The McGraw-Hill Companies 2004 398 CHAPTER 10 Structural Dynamics The amplitudes and phase angles are determined by applying the initial conditions which are u2 0 1 A 1 sin ộj A 2 sin Ộ2 22 73 0 2A 2 sin Ộ1 - 2 sin Ộ2 3 2 1 2 2 72 0 0 2 cos Ộ1 A 2 cos ộ2 22 73 0 0 2 A 2 cos Ộ1 - A 2 cos ộ2 22 The initial conditions produce a system of four algebraic equations in the four unknowns A 1 A 2 Ộ1 ộ2. Solution of the equations is not trivial owing to the presence 22 of the trigonometric functions. Letting P A 1 sin Ộ1 and Q A 2 sin ộ2 the displace-2 1 2 2 ment initial condition equations become P Q 1 2P - Q which are readily solved to obtain P A 2 sin Ộ1 and Q A 2 sin ộ2 2 1 2 2 Similarly setting R aTcos Ộ1 and 5 A 2 sin ộ2 the initial velocity equations are 22 R 0 2 R - 5 0 representing a homogeneous system in the variables R and 5. Nontrivial solutions exist only if the determinant of the coefficient matrix is zero. In this case the determinant is not zero as may easily be verified by direct computation. There are no nontrivial solutions hence R 5 0. Based on physical argument the amplitudes cannot be zero so we must conclude that cos Ộ1 cos ộ2 0 Ộ1 ộ2 W 2. It follows that the sine function of the phase angles have unity value hence A 2 and A 2 . Substituting the amplitudes into the general solution form while noting that sin tót 2 cos Mt the free-vibration response of each mass is u2 t U3 t cos - cos The displacement response of each mass is seen to be a combination of motions corresponding to the natural circular frequencies of the system. Such a phenomenon is characteristic of vibrating structural systems. All the natural modes of vibration participate in the general motion of a structure. Many-Degrees-of-Freedom Systems As illustrated by the system of two .