Absolutely, let's break down and explain the question step by step! --- ## **The Problem Overview** You have a radial saw base with: - **Mass, \( M = 73.16 \) kg** - Driven by a motor that causes it to vibrate due to imbalance. - The motor applies a **force, \( F = 13 \) N** at **180 rev/min**. - The base is supported by table legs with a **stiffness, \( k = 2600 \) N/m**. - **Maximum allowed deflection** is \( x_{max} = 0.2 \) cm (convert to meters: 0.002 m). - **Goal:** Add a vibration absorber to eliminate oscillations (deflection = 0 at operating speed). - **System is undamped.** --- ## **Questions to Solve** 1. **Natural frequency of the base (\( \omega_n \)) without absorber** 2. **Determine the mass and stiffness of the vibration absorber** --- ### **Step 1: Natural Frequency of the Base (Undamped System)** The formula for natural frequency (\( \omega_n \)) of a mass-spring system is: \[ \omega_n = \sqrt{\frac{k}{M}} \] Where: - \( k = 2600 \) N/m (stiffness) - \( M = 73.16 \) kg (mass) **Plug in the values:** \[ \omega_n = \sqrt{\frac{2600}{73.16}} \] \[ \omega_n = \sqrt{35.53} \approx 5.96 \ \text{rad/s} \] --- ### **Step 2: Vibration Absorber Design** #### **a) Find the excitation frequency** The **motor speed** is **180 rev/min**. Convert this to radians per second: \[ \text{Frequency in Hz} = \frac{180}{60} = 3 \ \text{Hz} \] \[ \omega = 2\pi f = 2\pi \times 3 = 6\pi \approx 18.85 \ \text{rad/s} \] #### **b) Principle of Vibration Absorber** A vibration absorber is a small mass-spring system attached to the main system. To eliminate the vibration at the excitation frequency, **the absorber's natural frequency should be equal to the excitation frequency**: \[ \omega_a = \omega \] \[ \omega_a = \sqrt{\frac{k_a}{m_a}} = 18.85 \ \text{rad/s} \] You have **two unknowns**: absorber mass (\( m_a \)), and absorber stiffness (\( k_a \)). You can choose \( m_a \) first, then solve for \( k_a \): \[ k_a = m_a \omega^2 \] #### **c) Deflection Requirement** The **requirement is that deflection of the base = 0** with absorber attached. This is achieved by tuning the absorber's natural frequency to exactly match the excitation frequency. --- ### **Summary of Steps** **1. Calculate natural frequency of the base:** \[ \omega_n = \sqrt{\frac{k}{M}} \] **2. Calculate excitation frequency:** \[ f = 3 \ \text{Hz} \] \[ \omega = 18.85 \ \text{rad/s} \] **3. Choose absorber mass (\( m_a \)), calculate absorber stiffness (\( k_a \)):** \[ k_a = m_a \omega^2 \] **4. To get exact values:** Usually, \( m_a \) is chosen as a small fraction of \( M \) (e.g., 5-15%), but you can set it as a variable or check if the deflection limit gives you a minimum \( m_a \). --- ### **If You Need to Calculate Minimum Absorber Mass:** If you want **zero deflection** at the operating frequency, the mass and stiffness of the absorber are related by the tuning condition. You can select \( m_a \) (e.g., 5 kg, 10 kg), then compute \( k_a \). --- ## **Let Me Know** Would you like to see the full calculation with a chosen value for \( m_a \), or do you want to know more about the theory behind vibration absorbers? Let me know how detailed you want the explanation!