Ronald J. Anderson - Introduction to Mechanical Vibrations

5 1.5 The figure shows a cart of mass that is attached to the ground at point by a spring of stiffness and a damper with damping coefficient . The motion of the cart is forced by a harmonic motion at the end of another damper with coefficient .Write the equation of motion for the system using as the degree of freedom. Do this once using Newton's Laws and again using Lagrange's Equation and confirm that you get the same result from each method.Figure E1.5

6 1.6 The figure shows a thin, uniform rod with mass = and length = that has been released from rest where . The rod rotates about a frictionless pin at .Figure E1.6Write the nonlinear equation of motion for the rod using as the DOF.Derive the equilibrium condition.What are the two equilibrium values of ?Linearize the equation of motion about each of the equilibrium states and compare the two differential equations. The result should be two, linear, second‐order, homogeneous differential equations that differ only in the sign of the coefficient multiplying . This coefficient is the “stiffness” and having a negative stiffness makes an equilibrium state unstable. Does your unstable state make physical sense?

1 1The normal force is what determines the torque that some external mechanism must apply to the wire in order to enforce the constraint that it rotate with constant angular velocity. The analysis here, like many dynamic analyses, assumes that torque is available and simply works with the constraint on the angular velocity.

2 2Joseph‐Louis Lagrange (1736–1813), an Italian/French mathematician, is well known for his work on calculus of variations, dynamics, and fluid mechanics. In 1788 Lagrange published the Mécanique Analytique summarizing all the work done in the field of mechanics since the time of Newton, thereby transforming mechanics into a branch of mathematical analysis.

3 3Note the difference between dimensions and units. Dimensions refer to physical characteristics such as mass, length, or time. Units refer to the system of measurement we use to substitute numbers into an equation. Examples are kilograms for mass, feet for length, and minutes for time.

4 4To see that this is the case, first divide both sides of Equation 1.30 by , then let go to zero and note that the left‐hand sides become , , and respectively. On the right‐hand side, becomes for all i. Finally, take the partial derivatives with respect to and the expressions in Equation 1.38result.

5 5The definition of the often‐quoted simple pendulum is that it has a massless rigid rod supporting a point mass. The rod is free to swing in a plane about the frictionless point where it is connected to the ground. The only external force is that due to gravity.

6 6All of these equilibrium states have very interesting stability characteristics but considering stability is outside the scope of what we are doing here. It is sufficient for our purposes to say, without proof, that the equilibrium state between and is stable.

7 7While the “sum of angles formulae” may be “well known”, they are not easily remembered. However, they can be quickly derived using Euler's equationWe write the equation twice, once for the angle and again for the angle , givingWe then multiply these two equations together giving, on the left‐hand side,and, on the right‐hand side,Equating the real and imaginary parts of these two results gives the “sum of angles” formulae we couldn't remember

8 8In the case of an angle of ten degrees, for example, we convert the angle to 0.174533 radians. The percentage errors using the linear approximations are only 0.5% on the sine and 1.5% on the cosine. Ten degrees of rotation is a very large angle in the world of vibrations. We are typically looking at fractions of a degree where the linear approximations are very accurate.

9 9For those readers who already know about the standard form of these equations and are worried about the consequences of the term in square brackets being negative, I suggest you take time to consider what value must have for this term to be negative.

10 10All of the analysis here has assumed that the motion of the system can be described with a single degree of freedom and therefore a single equation of motion. We will soon see that this is not the case but that the material and methods presented here still apply.

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