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**Dynamics Chapter 8. Simple Harmonic Motion and Vibeology

Recommended Reading: 【Physics】 Physics Table of Contents

1. Vibeology

2. Type 1. Simple Harmonic Motion

3. Type 2. Damped Oscillation

4. Type 3. Forced Oscillation

5. Mechanical Impedance

6. Mechanical Resonance

7. Superposition

a. Simple Harmonic Motion Experiment

1. Vibeology

⑴ Definition: The mechanical motion of an object exhibiting wave-like properties

2. Type 1. Simple Harmonic Motion** (Simple Undamped Oscillation, Simple Harmonic Motion)

Figure 1. Simple Harmonic Motion Diagram

⑴ Overview: Constant period

⑵ Cause: When restoring force and inertia act together in a system, simple harmonic motion occurs

① Restoring force: A force proportional to the displacement change

② Typically obtained through differential equations like y” + Ky = 0, K > 0

○ The term y” arises from F = ma

○ The Ky term appears because the restoring force is proportional to displacement change

③ Total energy due to restoring force

○ E = K + U

○ E = ½ kA², where A is the amplitude (maximum displacement)

④ Potential energy due to restoring force

○ Relationship between potential and force: F = -∇U

○ Since restoring force F ∝ x, U ∝ x²

○ Potential energy due to restoring force is lowest at the equilibrium point and increases as displacement changes

○ Stability classification

○ Stable equilibrium: U” > 0 (convex downward), U’ (x=0) = 0

○ Unstable equilibrium: U” < 0 (convex upward), U’ (x=0) = 0

⑤ Nonlinear effect

○ When the displacement from equilibrium is large enough that the restoring force is no longer proportional to displacement

○ Leads to modified simple harmonic motion

⑶ System Definition

① Solution to the differential equation

② Frequency Domain Conversion

⑷ Example 1. Simple Harmonic Motion Under a Spring

① Definition: A mass undergoing horizontal oscillation on a massless spring

② Elastic Force(Elastic Force): A force that restores an object to its original state due to atomic repulsion

○ Elasticity: The tendency of an object to return to its original shape when deformed

○ Elastic body: A body with elasticity (e.g., springs, rods, rubber used in keyboards)

○ Elastic force: A measure of an elastic body’s restoring force

○ Elastic limit: The point beyond which an elastic body cannot return to its original state

○ Hooke’s Law: The extension of a spring is proportional to the applied force

○ Equation: F ∝ x (deformation length) → F = -kx (where k is the elastic constant)

○ The negative sign indicates the force is in the opposite direction of elongation

○ Elastic Constant (Spring Constant): The force required to extend a spring by 1 m

○ Coil spring

○ Overlapping plate spring

○ Compliance: The inverse of the spring constant

○ Change in spring constant for series connection: Springs connected in series stretch more easily, reducing the spring constant

○ Change in spring constant for parallel connection: Springs connected in parallel require more force, increasing the spring constant

③ Simple Harmonic Motion by Elastic Force: Establishing equations using external force = restoring force = elastic force, then solving the second-order differential equation

④ Conclusion: The period T of a spring depends only on mass m and elastic constant k

⑸ Example 2. Simple Pendulum

① A mass suspended by a massless string oscillating with small amplitude

② The period T of the pendulum depends only on the string length ℓ

③ Physical pendulum: When a rigid body undergoes pendulum motion about a pivot in a gravitational field

④ Here, I is the moment of inertia about the pivot, and h is the distance from the pivot to the center of mass

⑹ Example 3. Simple Harmonic Motion of a Mass Suspended on a Spring in Gravity (Neglecting Spring Mass)

① Given a mass m and the gravity-induced extension x (which may be negative),

○ Φ: Initial phase

○ x_m: Amplitude (maximum displacement)

② Initial Condition: Assuming v = x = 0 at t = 0

③ Tip 1. The center of oscillation is where elastic force and gravity balance

④ Tip 2. The amplitude is determined by the initial conditions: The distance from the initial position to the oscillation center

⑤ Tip 3. The period: Independent of initial speed, extension length, and gravitational acceleration

⑺ Example 4. Simple Harmonic Motion of a Mass Suspended on a Spring in Gravity (Considering Spring Mass) (ref)

Figure 2. Considering the Mass of the Spring

① Condition 1. Mass of the bob M, mass of the spring m, spring constant k, spring length L, distance from attachment point L

② Condition 2. The spring is uniform, L ≫ extended length x

③ Under force equilibrium, the extended length x(y) at a point y from the top is

④ This equation implies a varying spring constant along the position, treating the spring mass as distributed

⑤ Finding the average spring constant

⑥ For m ≪ M, using the Taylor series expansion of ln(1 + x),

⑦ The approximate period formula follows, assuming (1 + x)-1 ≒ 1 - x

⑧ A more precise calculation is highly complex

⑻ Example 5. Pendulum Wave

① Definition: 15 pendulums moving in and out of phase over a 60-second cycle, creating a wave-like motion (ref)

② Design: Fine-tuning the pendulum lengths so that within 60 seconds, the 1st to 15th pendulums oscillate 51 to 65 times, respectively

⑼ Example 6. Exact Solution for Pendulum Motion

① Premises

○ Situation: Free undamped simple pendulum motion

○ Governing equation

○ Initial conditions

○ Approximation of the governing equation (θ ≪ 1)

○ Approximate general solution (θ ≪ 1)

○ Approximate particular solution (θ ≪ 1)

② Exact Solution

○ Transformation

○ Trigonometric relations

○ 1^st substitution

○ 2^nd substitution: ω0 comes from the governing equation, independent of angular frequency

○ Limitation for z

○ Exact solution for τ

③ Jacobi Elliptic Function

○ Elliptic integral

○ Definition of Jacobi elliptic function sn

○ Expression of τ using elliptic integral

○ Period formula involving K(k)

○ Expression of sn in terms of τ

○ Conclusion

⑽ Example 7. Convergence of the Exact Solution of Pendulum Motion under Air Resistance (ref)

① Problem: Let u(t) be the solution of the following differential equation. Show that the functions u(t) and u’(t) are bounded for t > 0. Also, determine the limit lim_t→∞ u(t).

② Solution:

First, define v(t) := u’(t). Then, given that (u(0), v(0)) = (1, 0), the following equation holds:

Thus, we obtain: -et ≤

et v(t)

’ = et sin u(t) ≤ et.

By integrating, we get:

v

≤ 1 - e-t ≤ 1. Next, define w(t) := 2 cos u(t) + (v(t))2.

Then, we compute: w’ = -2u’ sin u + 2vv’ = 2v sin u + 2v(-v + sin u) = -2v².

Since w(t) is a decreasing function, it follows that: cos u(t) ≤ cos u(t) + (v(t))² / 2 ≤ cos u(0) + (v(0))² / 2 = cos 1.

Using the continuity of u(t), we obtain: 1 ≤ u(t) ≤ 2π-1

Since w(t) is a decreasing function and satisfies w(t) ≥ -2, the limit lim_t→∞ w(t) = β exists, and we have:

Now, assume β > 2.

Then, for any ε > 0, there exists some t0 such that for all t > t0, cos u(t) > β / 2 - ε,

v(t)

< ε

If we assume β / 2 - ε > -1 for sufficiently small ε > 0, then two possible cases arise:

Case 1: This contradicts the boundedness of v for sufficiently small ε > 0.

Case 2: Similar contradiction follows.

Thus, we conclude that: lim_t→∞ cos u(t) = -1 and lim_t→∞ u(t) = π.

⑾ Example 8. Demonstrating Circular Motion as Simple Harmonic Motion

⑿ Example 9. LC Circuit

Figure 3. Oscillation in an LC Circuit

Figure 4. Analysis of an LC Circuit

⒀ Example 10. Atomic Oscillations

3. Type 2. Damped Oscillation

Figure 5. Damped Oscillation Diagram

⑴ Governing Equation

⑵ Frequency Domain Conversion

⑶ Introduction of β and Classification of Damped Oscillation

① β² - ω₀² > 0: Overdamped (heavy damping)

② β² - ω₀² = 0: Critically damped

③ β² - ω₀² < 0: Underdamped (light damping)

⑷ Generally, β < ω₀ holds

4. Type 3. Forced Oscillation

Figure 6. Forced Oscillation Diagram

⑴ Governing Equation

⑵ When f_e is a trigonometric function

① Oscillation = Natural Vibration + Steady-State Vibration

② Natural Vibration = Homogeneous Solution = Complementary Solution = Transient Solution

○ Due to e^-βt, it converges to 0 over time

③ Steady-State Vibration = Particular Solution

○ Typically, only the particular solution is of interest

○ Assumed as complex harmonic motion

○ ω ≪ ω₀

○ ω = ω₀

○ ω ≫ ω₀

5. Mechanical Impedance

⑴ Definition

① Defined as complex driving force ÷ complex velocity at the driving point

② Concept of force magnitude required to obtain velocity

⑵ SDOF (Single Degree of Freedom) System

① Z_m: Inherent property of material, used to predict response

② R_m: Mechanical resistance, representing energy loss

③ X_m: Mechanical reactance, representing energy storage

④ If impedance is known, the system response can be determined

⑶ When ω ≪ ω₀: Z_m* = -js / ω

① Spring-like (stiffness-like) impedance

② Negative imaginary component

③ Inversely proportional to frequency

⑷ When ω = ω₀: Z_m* = R_m

⑸ When ω ≫ ω₀: Z_m* = jωm

① Mass-like impedance

② Positive imaginary component

③ Linearly proportional to frequency

⑹ Imaginary Impedance: When R_m = 0

① Force and velocity are out of phase

② No power is required to drive the system

○ Since R_m = 0, no energy is dissipated

6. Mechanical Resonance

⑴ Definition: The frequency at which the reactance component X_m becomes zero

⑵ The vibration frequency at which the imaginary part of impedance is zero

7. Superposition

Figure 7. Superposition Diagram

⑴ Assumption 1. Small Amplitude Response

⑵ Assumption 2. The System is Linear

⑶ Mathematical Formulation

Input: 2019.04.09 08:59

Modified: 2023.10.07 17:04